Segmented Hybrid Impedance Control for Hyper-Redundant Space Manipulators

Abstract

Hyper-redundant space manipulators (HRSMs), with their extensive degrees of freedom, offer a promising solution for complex space operations such as on-orbit assembly and manipulation of non-cooperative objects. A critical challenge lies in achieving stable and effective grasping configurations, particularly when dealing with irregularly shaped objects in microgravity. This study addresses these challenges by developing a segmented hybrid impedance control architecture tailored to multi-point contact scenarios. The proposed framework reduces the contact forces and enhances object manipulation, enabling the secure handling of irregular objects and improving operational reliability. Numerical simulations demonstrate significant reductions in the contact forces during initial engagements, ensuring stable grasping and effective force regulation. The approach also enables precise trajectory tracking, robust collision avoidance, and resilience to external disturbances. The complete non-linear dynamics of the HRSM system are derived using the Kane method, incorporating both the free-space and constrained motion phases. These results highlight the practical capabilities of HRSM systems, including their potential to grasp and manipulate obstacles effectively, paving the way for applications in autonomous on-orbit servicing and assembly tasks. By integrating advanced control strategies and robust stability guarantees, this work provides a foundation for the deployment of HRSMs in real-world space operations, offering greater versatility and efficiency in complex environments.

1. Introduction

Space exploration depends on critical operations such as satellite inspection, refueling, repairs, orbital debris removal, and the construction and maintenance of orbital infrastructure. While astronauts have traditionally performed these tasks through spacewalks, such operations are inherently dangerous, requiring extensive planning and preparation. These constraints often make servicing missions prohibitively costly or impractical. Robotic servicing, particularly using hyper-redundant space manipulators (HRSMs), has emerged as a promising alternative due to the enhanced workspace and dexterity compared to traditional manipulators [1,2,3].

The historical approach to HRSMs has predominantly focused on developing methodologies to maintain environmental separation during operation. This paradigm is exemplified by Khatib’s pioneering work on real-time collision prevention for robotic systems, which introduced the artificial potential field methodology [4]. Further advancements came from Chirikjian and colleagues, who explored tunnel-constrained manipulator configurations through curvature function analysis [5]. The field’s theoretical foundation was enhanced by M. da Graca Marcos et al.’s investigation into fractional dynamics’ chaotic behavior using the Fourier and wavelet analytical techniques [6]. Contemporary contributions include Ni et al.’s development of synchronized trajectory planning for dual-arm space robotics [7], while C. Lauretti et al. introduced a geometric solution to the inverse kinematics challenges in hyper-redundant systems, specifically targeting real-time human-interactive control applications [8]. The emergence of artificial intelligence has catalyzed novel approaches to manipulation and obstacle avoidance, particularly through neural networks and evolutionary computing [9,10,11,12]. Notable recent developments include Zhai et al.’s adaptive neural synchronization framework for cooperative manipulation [13] and Ni et al.’s optimization-based approach to space robot trajectory planning, utilizing particle swarm optimization techniques [14].

However, our research reveals a paradigm shift. Controlled interaction between HRSMs and their surroundings can actually enhance operational capabilities rather than limit them. Environmental contact, when properly managed, can provide additional support points and stability, potentially improving the overall precision and load-bearing capacity of the system, as investigated by Luo et al. [15]. This approach is particularly relevant in scenarios where the hyper-redundant structure can leverage its extensive degrees of freedom to distribute the forces across multiple contact points, similar to how biological systems like snakes or tentacles utilize environmental interaction to enhance their mobility and manipulation capabilities. In this context, a hybrid impedance control with an object manipulation controller for HRSMs is developed.

Conventional control methods and recent research exhibit significant limitations when managing multiple contact points, as they are primarily designed for single-point contact scenarios. In the dynamic conditions of space operations, single-point contacts often prove insufficient for maintaining stability, leading to potential slippage, imbalance, or unwanted rotations. This limitation becomes particularly critical when intermediate joints inevitably interact with the environment, highlighting the need for more comprehensive control strategies. Multi-point contact systems have emerged as a more robust alternative, offering enhanced stability through even force distribution across objects and providing essential redundancy in grasp stability.

Multi-point contact impedance research in space manipulation has predominantly focused on the end-effector morphology. This focus emerged naturally from the challenge of grasping and manipulating objects in space, where end-effector-centric approaches have been highly investigated for both single- and multi-point contact scenarios. Conventional methodologies consistently address the contact problem through specialized end-effector designs [16], for example, as male–female connector interfaces [17], nozzle–cone configurations and using cooperative manipulators [18], which in a way limit the scope of possible interactions. Our research introduces a new approach by investigating a fundamentally different question: how can we control and leverage the entire segmented HRSM link structure to effectively manipulate irregular, non-cooperative objects in space environments?

Referring to the recent work of G. Xiong et al. [19], which represents one of the most closely related approaches to this research, the authors investigated the management of multiple external forces applied to redundant manipulators. Their study specifically explored the use of variable impedance control parameters to handle multi-point contacts in redundant manipulators. However, their approach, while comprehensive, lacks a mechanism for applying the desired forces that could enhance the stability in some tasks and ensure a firm grasp on uncooperative obstacles. In contrast, the present work introduces a segmented impedance control architecture designed to effectively manage multiple desired contact forces. This approach not only improves the stability of the manipulator during interactions with complex, uncooperative environments but also enables a hyper-redundant space manipulator (HRSM) to securely grasp and manipulate multiple objects through multi-point contacts for each.

The critical challenge in space manipulation centers on managing the contact dynamics [3]. Contact between rigid materials generates extremely high forces, with the transition between initial and permanent contact often exhibiting unstable, bouncy behavior. This instability is particularly problematic in space operations, where the object mass and inertia generate additional forces requiring careful regulation. The combination of contact instability, free-floating object unpredictability, microgravity conditions, and manipulator control errors often leads to system divergence into uncontrollable states. In this context, impedance control methodologies [20,21] show particular promise by replicating human-like muscle compliance and reducing the contact forces. Several researchers have made significant contributions to this field. Zhang et al. pioneered force feedback-based compliance control for manipulators [22], while Pathak et al. extended impedance control applications specifically for space manipulation scenarios [23]. Further advancements came from Wang et al., who adapted impedance control for non-cooperative target grasping [24], and Huang et al., who developed adaptive control strategies for target acquisition [25]. Notably, Uyama et al. investigated impedance control methods specifically for free-floating space manipulators engaging with non-cooperative targets [26]. However, traditional implementations of these methodologies still face challenges in effectively managing simultaneous multiple contact points, necessitating further development of more sophisticated control frameworks.

To address these fundamental challenges, we propose a novel bio-inspired approach incorporating winding or wrapping around non-cooperative objects. This solution provides comprehensive object securing, enhancing the stability and control during interaction. Furthermore, we introduce a segmented control approach for HRSMs, dividing the manipulator into independently controlled segments to simplify the kinematic and dynamic complexities while improving the maneuverability and responsiveness.

The key contributions of this work are as follows:

• Development of a comprehensive segmented impedance control framework enabling coordinated multi-point contact, demonstrating superior compliance and stability compared to traditional hybrid position/force control methods.
• Introduction of an integrated object control strategy that actively manages the forces arising from the object mass and motion during manipulation, ensuring stable control under varying inertial conditions.
• Presentation of a complete control architecture enabling HRSMs to wrap around, stabilize, and manipulate free-floating objects.

Through mathematical modeling using Kane’s method, we derive the complete dynamic equations governing the HRSM system. This theoretical foundation supports our control design and stability analysis, ensuring robust performance across various manipulation scenarios.

Using Figure 1 as an illustration, the subsequent sections of this paper delve into the development and application of segmented hybrid impedance control. Here, the manipulator is tasked with tracking the force at the contact point of the second segment and exhibiting impedance behavior using joint 15 to joint 24. Conversely, joint 1 to joint 14 of the manipulator are solely focused on achieving object position control, obstacle avoidance and self-avoidance, thereby allowing meticulous movement of the object.

The structure of this paper is as follows. Section 2 presents the system design of the segmented hybrid impedance control and outlines the system description of the HRSM. In Section 3, the dynamic equations of the HRSM are derived using Kane’s method, alongside an introduction to a contact collision model. Section 4 introduces the segmented hybrid impedance control strategy, object stabilization system and real-time collision avoidance. Section 5 demonstrates the numerical simulation results and validates the goals of the control strategy concerning the contact state, behavior of intermediate joints, object behavior and collision avoidance. Finally, the concluding section summarizes the paper and draws conclusions.

2. System Design

The system is considered to be a rigid multi-body system as the links’ and the joints’ flexibility is not considered. As shown in Figure 2 the set-up involves a hyper-redundant manipulator named M, alongside two rigid bodies, B1 and B2. B1 represent the satellite or the base and B2 represent a free-floating object. To keep track of the system’s movements, we use an inertial frame denoted as $ℱ$_e. Within this system, there are a total of 24 rigid bodies and 24 joints. Although these bodies and joints can be arbitrarily labeled, for purposes of clarity, each joint is assigned the same number as its connected outboard body (as shown in Figure 2). Each body Bj (where j = 1, 2, …, 24) is equipped with its own fixed frame $ℱ$j, aiding in the analysis of its movement and orientation.

Given that the mass of spacecraft bodies is generally large, we adopt the assumption of a fixed base position; however, as explained later, the attitude of the base is controlled. This rationale allows us to simplify the system analysis by assuming the base does not significantly move or deform, thereby concentrating on the dynamics and control of the manipulator itself. Under this premise, there are N revolute joints in the system. We designate the angular velocity of each revolute joint as the generalized velocity, described in the local coordinate systems of each joint. Consequently, the generalized velocity of the hyper-redundant space robotic manipulator system can be expressed as follows:

$$\mathrm{u}=\left[\begin{array}{c}{\dot{\mathsf{\theta }}}_1\\ ⋮\\ {\dot{\mathsf{\theta }}}_{\mathrm{n}}\end{array}\right],$$

(1)

For a revolute joint ${\dot{\mathsf{\theta }}}_{\mathrm{j}}={\dot{\mathrm{q}}}_{\mathrm{i}}$, where ${\dot{\mathsf{\theta }}}_{\mathrm{j}}$ is the relative rotational angle, the generalized velocity according to Maggi–Kane’s method can be written as follows:

$$\dot{\mathrm{q}}=[{\dot{\mathrm{q}}}_1^{\mathrm{T}},{\dot{\mathrm{q}}}_2^{\mathrm{T}},\dots ,{\dot{\mathrm{q}}}_{24}^{\mathrm{T}}{]}^{\mathrm{T}},$$

(2)

where ${\dot{\mathrm{q}}}_{\mathrm{i}}$ is the generalized speed of Bj. It describes the relative motion between Bj and its inboard body Bc(j), which is located near the base body and is connected to Bj by joint hj. The superscript “T” denotes the transpose of a matrix.

3. Equations of Motion

The equations of motion are constructed using Maggi–Kane’s method, represented in a matrix format. Hu et al. expanded the concepts of “partial velocity matrix” and “partial angular velocity matrix”, enhancing the efficiency of the algorithm and making it more adaptable for computer-based solutions [27]. Furthermore, a force model for vertical contact and a friction force model are developed, both stemming from a collision detection algorithm. Consequently, this work introduces collision detection rooted in computer graphics as an integral part of the approach.

3.1. Kane’s Formulations

Based on Kane’s method, the HRSM dynamic equation is:

$${\mathrm{F}}_{\mathrm{i}}^{\mathrm{I}}+{\mathrm{F}}_{\mathrm{i}}^{\mathrm{A}}=0,$$

(3)

where (i = 1, 2…, N), the generalized inertial force and generalized active force of the i-th body are denoted as ${\mathrm{F}}_{\mathrm{i}}^{\mathrm{I}}$ and ${\mathrm{F}}_{\mathrm{i}}^{\mathrm{A}}$, respectively, and N is the number of generalized speeds written in column matrix form.

The dynamic equation can be written as follows:

$$\mathrm{M}\dot{\mathrm{q}}+{\mathrm{F}}_{\mathrm{t}}^{\mathrm{I}}={\mathrm{F}}^{\mathrm{A}},$$

(4)

where M is the mass matrix, $\dot{\mathrm{q}}$ is the generalized speed, ${\mathrm{F}}_{\mathrm{t}}^{\mathrm{I}}$ is the generalized active force, including the vertical contact force ${\mathrm{f}}_{\mathrm{n}}$, friction force ${\mathrm{f}}_{\mathrm{t}}$ and control torque ${\mathsf{\Gamma }}_{\mathrm{M}}$, and ${\mathrm{F}}^{\mathrm{A}}$ is the generalized active force:

$$\mathrm{M}={\sum }_{\mathrm{j}=1}^{\mathrm{N}}{\mathrm{M}}_{\mathrm{j}},$$

(5)

$${\mathrm{F}}_{\mathrm{t}}^{\mathrm{I}}={\sum }_{\mathrm{j}=1}^{\mathrm{N}}{\mathrm{F}}_{\mathrm{j}\mathrm{t}}^{\mathrm{I}},$$

(6)

Making the assumption that the system has external forces and external moments represented by S and T, respectively, the generalized active force can be calculated by:

$${\mathrm{F}}^{\mathrm{A}}={\sum }_{\mathrm{i}=1}^{\mathrm{N}}({\sum }_{\mathrm{s}=1}^{\mathrm{S}}{}_{\mathrm{i}}{}^{\mathrm{p}}\mathrm{v}_{\mathrm{s}}{\mathrm{F}}_{\mathrm{s}}+{\sum }_{\mathrm{t}=1}^{\mathrm{T}}{}_{\mathrm{i}}{}^{\mathrm{p}}\omega _{\mathrm{t}}{\mathrm{T}}_{\mathrm{t}}),$$

(7)

where F_s and T_t are the external force and external moment, while ${}_{\mathrm{i}}{}^{\mathrm{p}}\mathrm{v}_{\mathrm{s}}$ and ${}_{\mathrm{i}}{}^{\mathrm{p}}\omega _{\mathrm{t}}$ are the partial velocity matrix and the partial angular velocity matrix. This equation aims to derive the generalized active force for each order of the generalized rate of the external forces and moments acting on the system, and subsequently, sum them up sequentially. Then we obtain (4).

3.2. Contact Model

In order to articulate the force applied where the manipulator interacts with the truss structure, we establish models for both the vertical contact force and friction. The contact model used in this manuscript follows the same logic used in reference [15]. Two algorithms were developed to address the detection and analysis of collisions between a link and an obstacle, both modeled as cylinders. The first algorithm based on computer-graphics-based methods identifies whether a collision has occurred, determines the collision point, classifies the collision as either dynamic or static, and assesses the direction of the resulting forces using a friction model.

The closest distance between the two cylindrical axis segments, as shown in Figure 3, ${\mathrm{P}}_{\mathrm{A}1}{\mathrm{P}}_{\mathrm{A}2}$ and ${\mathrm{P}}_{\mathrm{B}1}{\mathrm{P}}_{\mathrm{B}2}$ is determined by:

$${‖{\mathrm{l}}_{\mathrm{A}\mathrm{B}}‖}^2={({\mathrm{P}}_{\mathrm{A}1}-{\mathrm{P}}_{\mathrm{B}1}+{\mathrm{v}}_{\mathrm{A}0}\left({\mathrm{P}}_{\mathrm{A}2}-{\mathrm{P}}_{\mathrm{A}1}\right)-{\mathrm{v}}_{\mathrm{B}0}\left({\mathrm{P}}_{\mathrm{B}2}-{\mathrm{P}}_{\mathrm{B}1}\right))}^2,$$

(8)

where the coordinates of ${\mathrm{P}}_{\mathrm{A}}$ and ${\mathrm{P}}_{\mathrm{B}}$ are calculated as ${\mathrm{P}}_{\mathrm{A}}={\mathrm{P}}_{\mathrm{A}1}+{\mathrm{v}}_{\mathrm{A}0}({\mathrm{P}}_{\mathrm{A}2}-{\mathrm{P}}_{\mathrm{A}1})$, ${\mathrm{P}}_{\mathrm{B}}={\mathrm{P}}_{\mathrm{B}1}+{\mathrm{v}}_{\mathrm{B}0}({\mathrm{P}}_{\mathrm{B}2}-{\mathrm{P}}_{\mathrm{B}1})$; while ${\mathrm{v}}_{\mathrm{A}0}$ and ${\mathrm{v}}_{\mathrm{B}0}$ are the parameters for evaluating the collision state. The outputs of this first algorithm are then utilized by a second algorithm, which calculates the normal and tangential forces for each link involved in the contact based on the Flores contact force model.

$${\mathrm{f}}_{\mathrm{n}}={\mathrm{K}}_{\mathrm{C}\mathrm{o}\mathrm{l}}{\mathsf{\delta }}^{{\mathrm{n}}_{\mathrm{C}\mathrm{o}\mathrm{l}}}+{\mathrm{X}}_{\mathrm{C}\mathrm{o}\mathrm{l}}{\mathsf{\delta }}^{{\mathrm{n}}_{\mathrm{C}\mathrm{o}\mathrm{l}}}\dot{\mathsf{\delta }},$$

(9)

The hysteresis damping factor is as follows:

$${\mathrm{X}}_{\mathrm{C}\mathrm{o}\mathrm{l}}={\displaystyle \frac{8\left(1-{\mathrm{C}}_{\mathrm{r}}\right){\mathrm{K}}_{\mathrm{C}\mathrm{o}\mathrm{l}}}{5{\mathrm{C}}_{\mathrm{r}}\dot{{\mathsf{\delta }}^{(-)}}}},$$

(10)

where ${\mathrm{K}}_{\mathrm{c}\mathrm{o}\mathrm{l}}$ is the contact stiffness coefficient, $\mathsf{\delta }$ is the embedded depth of the two contact rigid bodies, and Cr is the restitution coefficient ${\dot{\mathsf{\delta }}}^{(-)}$ indicating the speed at the beginning of the collision.

4. System Control

4.1. Hybrid Impedance Control of Hyper-Redundant Space Manipulators

When a hyper-redundant space manipulator is tasked with wrapping around an external target, it encounters multiple collisions with the surrounding environment at various points. In such scenarios, relying solely on a position controller or a force/position hybrid controller may compromise the manipulator’s motion accuracy, as these methods are heavily dependent on precise trajectory planning and controller performance. Moreover, the manipulator must also stabilize the forces arising from the mass of the object itself during collisions, which can further complicate control. As discussed in previous work [5], employing a multi-point contact, segmental force/position hybrid control approach often results in repeated contacts and separations with the target, generating significant instantaneous forces. These forces, especially in the case of frequent collisions or interactions with stiff materials, can not only reduce the operational accuracy but also cause structural damage. Therefore, there is a pressing need to develop an active compliance control strategy that can effectively manage the additional forces from both the object’s mass and the external collisions, thereby enhancing the safety and reliability of the manipulator’s operations.

The concept involves integrating impedance control mechanisms into the joints that come into contact with external objects.

4.2. Impedance Control Principle for Robotic Manipulators

The impedance control method aims to impart specific mass–stiffness–damping characteristics to the manipulator when it interacts with the external environment. Within the control methods for the manipulator, using position control reflects the system’s high impedance characteristics, meaning that external force disturbances have minimal impact on the manipulator’s motion variations. Conversely, employing force control reflects the system’s low impedance characteristics, where changes in the force due to motion errors are minimal. However, when the manipulator links and the external objects possess significant stiffness, using only force/position hybrid control may result in substantial force tracking errors caused by minor motion variations in the force control direction.

By employing impedance control, it is possible to establish a dynamic relationship between the contact force and the motion of the manipulator arm in the joint space as follows:

$${\mathrm{M}}_{\mathrm{e}\mathrm{t}\mathrm{m}}{\dot{\mathrm{u}}}_{\mathrm{e}\mathrm{A}}+{\mathrm{C}}_{\mathrm{e}\mathrm{t}\mathrm{m}}{\mathrm{u}}_{\mathrm{e}\mathrm{A}}+{\mathrm{K}}_{\mathrm{e}\mathrm{t}\mathrm{m}}{\mathrm{X}}_{\mathrm{e}\mathrm{A}}={\mathrm{F}}_{\mathrm{e}\mathrm{t}\mathrm{m}}^{\mathrm{C}},$$

(11)

where ${\mathrm{F}}_{\mathrm{e}\mathrm{t}\mathrm{m}}^{\mathrm{C}}$ is the contact force on the end-effector of the manipulator; and ${\mathrm{M}}_{\mathrm{e}\mathrm{t}\mathrm{m}}$, ${\mathrm{C}}_{\mathrm{e}\mathrm{t}\mathrm{m}}$ and ${\mathrm{K}}_{\mathrm{e}\mathrm{t}\mathrm{m}}$ are positive definite parameter matrices, corresponding to the mass, damping and stiffness characteristics of the system, respectively. In this paper, we focus on the case where they are constant matrices. At the same time, ${\dot{\mathrm{u}}}_{\mathrm{e}\mathrm{A}}$, ${\mathrm{u}}_{\mathrm{e}\mathrm{A}}$ and ${\mathrm{X}}_{\mathrm{e}\mathrm{A}}$ can be written as the correction quantities of the corresponding parameters, that is, $\mathsf{\Delta }{\dot{\mathrm{u}}}_{\mathrm{e}\mathrm{A}}$, $\mathsf{\Delta }{\mathrm{u}}_{\mathrm{e}\mathrm{A}}$ and $\mathsf{\Delta }{\mathrm{X}}_{\mathrm{e}\mathrm{A}}$.

The impedance control equation is equally applicable in the joint space of the robotic arm, representing the external contact force in the joint space of the robotic arm as follows:

$${\mathrm{F}}_{\mathrm{G}\mathrm{C},\mathrm{q}}={\mathrm{J}}_{\mathrm{q}}^{\mathrm{T}}{\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}},$$

(12)

where ${\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ is the measured contact force. ${\mathrm{J}}_{\mathrm{q}}^{\mathrm{T}}$ is the transpose of a six-dimensional vector function, known as the Jacobian matrix. Then, the impedance control equation in the joint space can be written as follows:

$${\mathrm{M}}_{\mathrm{A}}\mathsf{\Delta }{\ddot{\mathrm{q}}}_{\mathrm{A}}+{\mathrm{C}}_{\mathrm{A}}\mathsf{\Delta }{\dot{\mathrm{q}}}_{\mathrm{A}}+{\mathrm{K}}_{\mathrm{A}}\mathsf{\Delta }{\mathrm{q}}_{\mathrm{A}}={\mathrm{F}}_{\mathrm{G}\mathrm{C},\mathrm{q}},$$

(13)

where $\mathsf{\Delta }{\ddot{\mathrm{q}}}_{\mathrm{A}}$, $\mathsf{\Delta }{\dot{\mathrm{q}}}_{\mathrm{A}}$ and $\mathsf{\Delta }{\mathrm{q}}_{\mathrm{A}}$ are the expected trajectory correction amounts of the manipulator joints. It is worth noting that the impedance control represented by (11) is a generalized impedance control, and its application mode can be adjusted in practical use depending on the control variables. In this research, the robotic arm engages in multi-point and multiple-contact collisions with external objects, requiring force measurement and tracking. Therefore, in this context, a position-based impedance control (admittance control) strategy is employed. This strategy provides the correction of the joint angle acceleration trajectories as follows:

$$\mathsf{\Delta }{\ddot{\mathrm{q}}}_{\mathrm{A}}={\mathrm{M}}_{\mathrm{A}}^{-1}\left({\mathrm{F}}_{\mathrm{G}\mathrm{C},\mathrm{q}}-{\mathrm{C}}_{\mathrm{A}}\mathsf{\Delta }{\dot{\mathrm{q}}}_{\mathrm{A}}-{\mathrm{K}}_{\mathrm{A}}\mathsf{\Delta }{\mathrm{q}}_{\mathrm{A}}\right),$$

(14)

Based on the motion of the end-effector actuator, the desired trajectory for the joint angle acceleration of the manipulator arm can be expressed as follows:

$${\dot{\mathrm{q}}}_{\mathrm{d},\mathsf{\theta }}={\mathrm{J}}_{\mathsf{\theta }}^{†}\left(\mathrm{q}\right)\left({\ddot{\mathrm{x}}}_{\mathrm{d}}-{\dot{\mathrm{J}}}_{\mathsf{\theta }}\left(\mathrm{q}\right)\dot{\mathrm{q}}\right),$$

(15)

The corrected desired trajectory for the manipulator arm joints is:

$$\left\{\begin{array}{c}{\ddot{\mathrm{q}}}_{\mathrm{A}}=\Delta {\ddot{\mathrm{q}}}_{\mathrm{A}}+{\ddot{\mathrm{q}}}_{\mathrm{d},\mathsf{\theta }\mathrm{I}}\\ {\dot{\mathrm{q}}}_{\mathrm{A}}=\Delta {\dot{\mathrm{q}}}_{\mathrm{A}}+{\dot{\mathrm{q}}}_{\mathrm{d},\mathsf{\theta }\mathrm{I}}\\ {\mathrm{q}}_{\mathrm{A}}=\Delta {\mathrm{q}}_{\mathrm{A}}+{\mathrm{q}}_{\mathrm{d},\mathsf{\theta }\mathrm{I}}\end{array}\right.,$$

(16)

The structure of the position-based impedance controller (admittance controller) is shown in Figure 4. In this paper, the hyper-redundant space manipulator adopts a segmented approach in the joint space using a force/position hybrid control method. Therefore, it is necessary to solve the generalized active force based on the measured contact force values ${\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$, define the active force ${\mathrm{F}}_{\mathrm{G}\mathrm{C},\mathrm{q}}$, and then generate the desired trajectory correction quantities $\mathsf{\Delta }{\ddot{\mathrm{q}}}_{\mathrm{A}}$, $\mathsf{\Delta }{\dot{\mathrm{q}}}_{\mathrm{A}}$ and $\mathsf{\Delta }{\mathrm{q}}_{\mathrm{A}}$.

4.3. Hybrid Impedance Control Principle for Robotic Manipulators

The force/position hybrid control method aims to separate subspaces, allowing simultaneous tracking of the intended trajectories and the desired contact force magnitude. However, its force control aspect typically employs conventional PI control, which can introduce considerable stiffness to the robotic arm. This method relies on error signals in the force or position to generate control inputs, striving to reach the intended values. In environments with high uncertainties where suitable reference trajectories and force signals are unavailable, employing a force/position hybrid controller may still lead to substantial undesired contact forces.

Impedance control offers the robotic arm behavior akin to a mass–spring–damper system, adapting interaction forces when uncertainties in contact point positions and environmental characteristics arise. However, this control scheme relies on the dynamic correlation between force and position to regulate the contact forces and lacks the ability to concurrently achieve both trajectory and contact force tracking.

In response to this limitation, a proposed solution in the existing literature introduces a hybrid impedance control method that merges impedance control with force/position hybrid control [21,22]. This method retains the subdivision of subspaces, maintaining trajectory tracking within the position control subspace. The robotic arm’s trajectory is updated using the impedance model. Within the force control subspace, the desired inertia and damping are introduced, while setting the desired stiffness to zero ensures precise force control. The expected motion equation along the force control subspace can be written as follows:

$${\mathrm{M}}_{\mathrm{d}}\ddot{\mathrm{X}}+{\mathrm{B}}_{\mathrm{d}}\dot{\mathrm{X}}-{\mathrm{F}}_{\mathrm{d}}=-{\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}},$$

(17)

where ${\mathrm{M}}_{\mathrm{d}}$ and ${\mathrm{B}}_{\mathrm{d}}$ are the expected inertia and damping parameter matrices, both of which are positive definite matrices; ${\mathrm{F}}_{\mathrm{d}}$ is the expected contact force between the robotic arm and the external object; ${\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ is the actual contact forces; and $\ddot{\mathrm{X}}$ and $\dot{\mathrm{X}}$ are the acceleration and velocity trajectory vectors of the robotic arm. By adjusting the desired inertia and damping parameters, the system’s dynamic response can be modified. Additionally, because this model does not introduce stiffness parameters, it will not affect the desired position trajectory, allowing the system to complete trajectory tracking. When the robotic arm is not in contact with external objects, and when the actual contact force ${\mathrm{F}}_{\mathrm{e}}=0$, the reference acceleration trajectory can be obtained as follows:

$$\ddot{\mathrm{X}}={\mathrm{M}}_{\mathrm{d}}^{-1}\left(-{\mathrm{B}}_{\mathrm{d}}\dot{\mathrm{X}}+{\mathrm{F}}_{\mathrm{d}}\right),$$

(18)

At the same time, the external contact forces can be expressed in a form related to the contact material.

$${\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}}={\mathrm{M}}_{\mathrm{e}}\ddot{\mathrm{X}}+{\mathrm{B}}_{\mathrm{e}}\dot{\mathrm{X}}+{\mathrm{K}}_{\mathrm{e}}\mathrm{X},$$

(19)

where ${\mathrm{M}}_{\mathrm{e}}$, ${\mathrm{B}}_{\mathrm{e}}$ and ${\mathrm{K}}_{\mathrm{e}}$ are the actual inertia, damping and stiffness parameter matrices of the contact system, all of which are positive definite matrices. Combining (15) and (17), we can have the following:

$${\mathrm{F}}_{\mathrm{d}}=\left({\mathrm{M}}_{\mathrm{d}}+{\mathrm{M}}_{\mathrm{e}}\right)\ddot{\mathrm{X}}+\left({\mathrm{B}}_{\mathrm{d}}+{\mathrm{B}}_{\mathrm{e}}\right)\dot{\mathrm{X}}+{\mathrm{K}}_{\mathrm{e}}\mathrm{X},$$

(20)

From the above equation, it can be seen that impedance control in the force control direction can be achieved by modifying the desired inertia parameters ${\mathrm{M}}_{\mathrm{d}}$ and damping parameters ${\mathrm{B}}_{\mathrm{d}}$ to adjust the response. Based on the impedance control method mentioned above, it is possible to further design the control equations for hybrid impedance as follows:

$${\mathrm{M}}_{\mathrm{d}}\left(\ddot{\mathrm{X}}-\mathrm{S}{\ddot{\mathrm{X}}}_{\mathrm{d}}\right)+{\mathrm{B}}_{\mathrm{d}}\left(\dot{\mathrm{X}}-\mathrm{S}{\dot{\mathrm{X}}}_{\mathrm{d}}\right)+{\mathrm{K}}_{\mathrm{d}}\mathrm{S}\left(\mathrm{X}-{\mathrm{X}}_{\mathrm{d}}\right)-\left(\mathrm{I}-\mathrm{S}\right){\mathrm{F}}_{\mathrm{d}}=-{\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}},$$

(21)

where ${\ddot{\mathrm{X}}}_{\mathrm{d}}$, ${\dot{\mathrm{X}}}_{\mathrm{d}}$ and ${\mathrm{X}}_{\mathrm{d}}$ are the expected trajectories of the system; $\mathrm{S}$ and $\mathrm{I}-\mathrm{S}$ are the selection matrices of the position control subspace and force control subspace, respectively; and the diagonal elements are all zero or one. Solving (19), we can obtain the reference acceleration trajectory of the hybrid impedance control as follows:

$$\ddot{\mathrm{X}}={\mathrm{M}}_{\mathrm{d}}^{-1}[-{\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}}+\left(\mathrm{I}-\mathrm{S}\right){\mathrm{F}}_{\mathrm{d}}-{\mathrm{B}}_{\mathrm{d}}\left(\dot{\mathrm{X}}-\mathrm{S}{\dot{\mathrm{X}}}_{\mathrm{d}}\right)-{\mathrm{K}}_{\mathrm{d}}\mathrm{S}\left(\mathrm{X}-{\mathrm{X}}_{\mathrm{d}}\right)]+\mathrm{S}{\ddot{\mathrm{X}}}_{\mathrm{d}},$$

(22)

4.4. Segmented Impedance Controller

The hybrid impedance control method presented in (20) imparts impedance characteristics to the robotic arm in the end-effector Cartesian space. However, the segmented force/position hybrid control established in this paper formulates control equations in the joint space. Therefore, the rest of this section provides the hybrid impedance control equations in the joint space. Building on the research findings in reference [5], we develop segmented hybrid impedance control suitable for hyper-redundant robotic arms. The impedance behavior has proven to be necessary in our scenario due to the extensive number of contact points and the vibrations caused by other controllers. Based on equation ${\mathrm{F}}_{\mathrm{G}\mathrm{C},\mathrm{q}}={\mathrm{J}}_{\mathrm{q}}^{\mathrm{T}}{\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$, the generalized forces exerted by the hyper-redundant robotic arm on external objects can be derived as follows:

$${\mathrm{F}}_{\mathrm{G}\mathrm{C},\mathrm{q}}^{\sec 1}={\mathrm{J}}_{{\mathrm{q}}_{\sec 1}}^{\mathrm{T}}{\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}},$$

(23)

where ${\mathrm{J}}_{{\mathrm{q}}_{\sec 1}}^{\mathrm{T}}$ represents the constraint Jacobian matrix at the contact point of the robotic arm. Referring to (19), we can derive the joint-space hybrid impedance control equations as follows:

$${\mathrm{M}}_{\mathrm{A},\mathrm{d}}\left({\ddot{\mathrm{q}}}_{\mathrm{H},\mathrm{d}}-\mathrm{P}{\ddot{\mathrm{q}}}_{\mathrm{d}}\right)+{\mathrm{B}}_{\mathrm{A},\mathrm{d}}\left({\dot{\mathrm{q}}}_{\mathrm{H},\mathrm{d}}-\mathrm{P}{\dot{\mathrm{q}}}_{\mathrm{d}}\right)+{\mathrm{K}}_{\mathrm{A},\mathrm{d}}\mathrm{P}({\mathrm{q}}_{\mathrm{H},\mathrm{d}}-{\mathrm{q}}_{\mathrm{d}})-\left(\mathrm{I}-\mathrm{P}\right){\mathrm{F}}_{\mathrm{d}}=-{\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}},$$

(24)

where $\mathrm{P}=\mathrm{I}-{\mathrm{A}}^{\mathrm{T}}(\mathrm{A}{\mathrm{M}}^{-1}{\mathrm{A}}^{\mathrm{T}}{)}^{-1}\mathrm{A}{\mathrm{M}}^{-1}$, M $\in {\mathbb{R}}^{\mathrm{k}\times \mathrm{n}}$ is a mass matrix, A $\in {\mathbb{R}}^{\mathrm{k}\times \mathrm{n}}$ is a Pfaffian constraints matrix in joint space and I is the identity matrix. $\left(\mathrm{I}-\mathrm{P}\left({\mathrm{q}}_{\mathrm{H},\mathrm{d}}\right)\right)\in {\mathbb{R}}^{\mathrm{n}\times \mathrm{n}}$ is the subspace matrix of the force constraints, which projects the constrained forces acting on the end-effector of the manipulator arm to the subspace of force constraints. $\mathrm{P}\left({\mathrm{q}}_{\mathrm{H},\mathrm{d}}\right)\in {\mathbb{R}}^{\mathrm{n}\times \mathrm{n}}$, with rank $\mathrm{n}-{\mathrm{k}}_1$, is the subspace for position control. ${\mathrm{q}}_{\mathrm{d}}$ is the expected joint angle for the robotic arm, while ${\mathrm{M}}_{\mathrm{A},\mathrm{d}}$, ${\mathrm{B}}_{\mathrm{A},\mathrm{d}}$ and ${\mathrm{K}}_{\mathrm{A},\mathrm{d}}$ are the parameter matrices of system inertia, damping and stiffness, all of which are positive definite matrices.

In the segmented hybrid control task of the hyper-redundant robotic arm, the first segment of the arm contacts and leans on the target, with the control objective of ensuring that there is no relative motion between the first segment and the target. Therefore, the desired joint angles ${\mathrm{q}}_{\mathrm{d}}$, desired angular velocities ${\dot{\mathrm{q}}}_{\mathrm{d}}$, and desired angular accelerations ${\ddot{\mathrm{q}}}_{\mathrm{d}}$ are all zero matrices. Additionally, considering the precision requirements for position control of the first segment, only the cases with introduced inertia and damping are considered, making the impedance control not exhibit stiffness characteristics. This helps to avoid significant errors in the motion of the first segment of the arm due to trajectory corrections. As a result, the segmented hybrid impedance control equations can be expressed as follows:

$${\mathrm{M}}_{\mathrm{H},\mathrm{d}}{\ddot{\mathrm{q}}}_{\mathrm{H},\mathrm{d}}+{\mathrm{B}}_{\mathrm{H},\mathrm{d}}{\dot{\mathrm{q}}}_{\mathrm{H},\mathrm{d}}-\left(\mathrm{I}-\mathrm{P}\right){\mathrm{F}}_{\mathrm{d}}=-{\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}},$$

(25)

According to this equation, the reference trajectory for the robotic arm’s joint angle acceleration can be determined as follows:

$$\mathsf{\Delta }{\ddot{\mathrm{q}}}_{\mathrm{H},\mathrm{d}}={\mathrm{M}}_{\mathrm{H},\mathrm{d}}^{-1}[\left(\mathrm{I}-\mathrm{P}\right){\mathrm{F}}_{\mathrm{d}}-{\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}}-{\mathrm{B}}_{\mathrm{H},\mathrm{d}}\mathsf{\Delta }{\dot{\mathrm{q}}}_{\mathrm{H},\mathrm{d}}],$$

(26)

The corrected trajectory of the first segment of the robotic arm is:

$$\left\{\begin{array}{c}{\ddot{\mathrm{q}}}_{\mathrm{H}}=\Delta {\ddot{\mathrm{q}}}_{\mathrm{H},\mathrm{d}}+{\ddot{\mathrm{q}}}_{\mathrm{d}}\\ {\dot{\mathrm{q}}}_{\mathrm{H}}=\Delta {\dot{\mathrm{q}}}_{\mathrm{H},\mathrm{d}}+{\dot{\mathrm{q}}}_{\mathrm{d}}\end{array}\right.,$$

(27)

4.5. Segmented Hybrid Motion/Force Control

In the segmented force/position hybrid control method, it is required that the first segment of the hyper-redundant manipulator comes into contact with external objects and achieves stable leaning. This process necessitates attention to position control at the contact point to maintain the stability of the external forces according to the contact direction. Representing the contact force between the manipulator’s first segment and the target by ${\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$, the manipulator’s dynamic equation can be expressed as follows:

$$\mathrm{M}\left(\mathrm{q}\right)\ddot{\mathrm{q}}+\mathrm{C}(\mathrm{q},\dot{\mathrm{q}})+{\mathrm{J}}_{\mathrm{s}\mathrm{e}\mathrm{c}1}^{\mathrm{T}}\left(\mathrm{q}\right){\mathrm{F}}_{\mathrm{t}\mathrm{i}\mathrm{p}}=\mathsf{\tau },$$

(28)

In the force control problem, the velocity and acceleration of the manipulator arm according to the force tracking direction are typically neglected. Therefore, the dynamic equation of the manipulator arm can be simplified as follows:

$${\mathrm{J}}^{\mathrm{T}}\left(\mathrm{q}\right){\mathrm{F}}_{\mathrm{t}\mathrm{i}\mathrm{p}}=\mathsf{\tau },$$

(29)

The PI force controller equation can be expressed as follows:

$$\mathsf{\tau }={\mathrm{J}}_{\mathrm{s}\mathrm{e}\mathrm{c}1}^{\mathrm{T}}(\mathrm{q})({\mathrm{F}}_{\mathrm{d}}+{\mathrm{K}}_{\mathrm{f}\mathrm{p}}{\mathrm{F}}_{\mathrm{e}}+{\mathrm{K}}_{\mathrm{f}\mathrm{i}}\int {\mathrm{F}}_{\mathrm{e}}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}),$$

(30)

where ${\mathrm{F}}_{\mathrm{e}}={\mathrm{F}}_{\mathrm{d}}-{\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$, ${\mathrm{K}}_{\mathrm{f}\mathrm{p}}$ and ${\mathrm{K}}_{\mathrm{f}\mathrm{i}}$ are positive definite PI controller parameters. By combining the two previous equations, we obtain:

$${\mathrm{K}}_{\mathrm{f}\mathrm{p}}{\mathrm{F}}_{\mathrm{e}}+{\mathrm{K}}_{\mathrm{f}\mathrm{i}}\int {\mathrm{F}}_{\mathrm{e}}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}=0,$$

(31)

Based on this equation, it is evident that when both ${\mathrm{K}}_{\mathrm{f}\mathrm{p}}$ and ${\mathrm{K}}_{\mathrm{f}\mathrm{i}}$ are positive definite matrices, the force tracking error signal ${\mathrm{F}}_{\mathrm{e}}$ can converge to zero. However, if the manipulator arm has not experienced the anticipated contact collision with the target, the presence of force control can lead to significant velocity errors in the manipulator arm. Since the velocity term in typical force control tasks is relatively small, a positive definite parameter matrix ${\mathrm{K}}_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{p}}$ can be directly introduced to incorporate velocity damping terms. The modified control law becomes:

$${\mathsf{\tau }}^{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{e}}={\mathrm{J}}^{\mathrm{T}}(\mathrm{q})({\mathrm{F}}_{\mathrm{d}}+{\mathrm{K}}_{\mathrm{f}\mathrm{p}}{\mathrm{F}}_{\mathrm{e}}+{\mathrm{K}}_{\mathrm{f}\mathrm{i}}\int {\mathrm{F}}_{\mathrm{e}}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}-{\mathrm{K}}_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{p}}\nu ),$$

(32)

For the segmented position control of the manipulator, the control method employed is the traditional joint space position control. This involves establishing a correspondence between the workspace and the joint space using the Jacobian matrix. Based on the planned joint angle trajectory ${\dot{\mathrm{q}}}_{\mathrm{d}}$, the PD controller for position control can be expressed as follows:

$${\mathsf{\tau }}^{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}={\widehat{\mathrm{M}}}_{\sec 1}\left(\mathrm{q}\right)\left({\ddot{\mathrm{q}}}_{\mathrm{d}}\left(\mathrm{t}\right)+{\mathrm{K}}_{\mathrm{p}1}{\mathrm{q}}_{\mathrm{e}1}\left(\mathrm{t}\right)+{\mathrm{K}}_{\mathrm{d}1}{\dot{\mathrm{q}}}_{\mathrm{e}1}\left(\mathrm{t}\right)\right),$$

(33)

where ${\mathsf{\tau }}^{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}$ is the joint control torque of the manipulator, and ${\widehat{\mathrm{M}}}_{\sec 1}\left(\mathrm{q}\right)$ denotes an estimated value of the manipulator system’s mass matrix. ${\mathrm{K}}_{\mathrm{p}}$ and ${\mathrm{K}}_{\mathrm{d}}$ are coefficient matrices of the PD controller.

Introducing the principle of the force/position hybrid control method, considering the scenario where the manipulator arm joints come into contact with an object and needs to move on the surface of the target and assuming the manipulator make contact with rigid target. The constrained dynamic equations can be written as follows:

$$\mathrm{P}\left(\mathrm{q}\right)\mathsf{\tau }=\mathrm{P}\left(\mathrm{q}\right)\left(\mathrm{M}\right(\mathrm{q})\dot{\mathrm{q}}+\mathsf{\eta }(\mathrm{q},\dot{\mathrm{q}}\left)\right),$$

(34)

where $\mathrm{P}=\mathrm{I}-{\mathrm{A}}^{\mathrm{T}}(\mathrm{A}{\mathrm{M}}^{-1}{\mathrm{A}}^{\mathrm{T}}{)}^{-1}\mathrm{A}{\mathrm{M}}^{-1}$, M $\in {\mathbb{R}}^{\mathrm{k}\times \mathrm{n}}$ with a rank of ${\mathrm{K}}_1$ is the subspace matrix of the force constraints. This matrix projects the constrained forces applied to the end-effector of the manipulator arm onto the subspace of force constraints. In this direction, the manipulator tracks the desired magnitude of the constrained force.

Figure 5 illustrates the schematic diagram of the hybrid impedance controller for the manipulator, where the controller structure is divided into the inner loop of hybrid motion–force control and the outer loop of impedance control. The inner loop is a force/position hybrid control method, with the control signals from both controllers being allocated to different degrees of freedom directions by the mutually orthogonal subspaces $\mathrm{P}$ and $\left(\mathrm{I}-\mathrm{P}\right)$. In the outer loop of impedance control, the measured contact force ${\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ and the desired contact force ${\mathrm{F}}_{\mathrm{d}}$ are introduced into the impedance model. Subsequently, the concentrated contact force in the workspace is transformed into the generalized active force in the joint space. From the equation, it is evident that when the force tracking error ${\mathrm{F}}_{\mathrm{e}}=\left(\mathrm{I}-\mathrm{P}\right){\mathrm{F}}_{\mathrm{d}}-{\mathrm{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ is zero, the trajectory correction generated by the manipulator arm impedance model is zero.

4.6. Segmented Impedance Motion/Force Control

In the HRSM’s first segment operation task, if it has not yet made contact with external objects, there is no need to introduce impedance control. Instead, segmented force/position hybrid control is used to bring the first segment of the robotic arm closer to the target. Without collision, the segmented force/position hybrid control method is used, and there is no impedance in the controller. The control signal for the segmented manipulator arm is the same as the following equation:

$${\mathsf{\Gamma }}_{\sec 1}=\mathrm{P}\left(\mathrm{q}\right)({\tilde{\mathrm{M}}}_{\sec 1}\left(\mathrm{q}\right)({\ddot{\mathrm{q}}}_{\mathrm{d}}\left(\mathrm{t}\right)+{\mathrm{K}}_{\mathrm{p}1}{\mathrm{q}}_{\mathrm{e}1}+{\mathrm{K}}_{\mathrm{d}1}{\dot{\mathrm{q}}}_{\mathrm{e}1}))+(\mathrm{I}-\mathrm{P}\left(\mathrm{q}\right))({\mathrm{J}}_{\sec 1}^{\mathrm{T}}(\mathrm{q})({\mathrm{F}}_{\mathrm{d}}+{\mathrm{K}}_{\mathrm{f}\mathrm{p}}{\mathrm{F}}_{\mathrm{e}}+{\mathrm{K}}_{\mathrm{f}\mathrm{i}}\int {\mathrm{F}}_{\mathrm{e}}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}-{\mathrm{K}}_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{p}}\nu )),$$

(35)

When the HRSM link makes its first contact with the target object, the controller for the first segment of the robotic arm switches to the hybrid impedance control mode. This mode involves modifying the desired trajectory and continuing to detect and track the contact force. This allows smooth control of the robotic arm’s motion without changing the structure of the hybrid force/position controller (inner loop) shown in Figure 5.

After the arm collides with an external object, the desired corrections for angular velocity and angular acceleration are obtained based on the feedback of the contact force measurement and the designed segmented hybrid impedance control. At this point, the control signal for the first segment of the manipulator arm in joint space is given by:

$${\mathsf{\Gamma }}_{\sec 1}={\mathsf{\Gamma }}_{\sec 1}=\mathrm{P}\left(\mathrm{q}\right)({\tilde{\mathrm{M}}}_{\sec 1}\left(\mathrm{q}\right)({\ddot{\mathrm{q}}}_{\mathrm{H}}\left(\mathrm{t}\right)+{\mathrm{K}}_{\mathrm{p}1}\left({\mathrm{q}}_{\mathrm{d}}-\mathrm{q}\right)+{\mathrm{K}}_{\mathrm{d}1}\left({\dot{\mathrm{q}}}_{\mathrm{H}}-\dot{\mathrm{q}}\right)))+\left(\mathrm{I}-\mathrm{P}\left(\mathrm{q}\right)\right)({\mathrm{J}}_{\sec 1}^{\mathrm{T}}(\mathrm{q})({\mathrm{F}}_{\mathrm{d}}+{\mathrm{K}}_{\mathrm{f}\mathrm{p}}{\mathrm{F}}_{\mathrm{e}}+{\mathrm{K}}_{\mathrm{f}\mathrm{i}}\int {\mathrm{F}}_{\mathrm{e}}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}-{\mathrm{K}}_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{p}}\nu )),$$

(36)

where ${\ddot{\mathrm{q}}}_{\mathrm{H}}$ and ${\dot{\mathrm{q}}}_{\mathrm{H}}$ are the expected angular acceleration and angular velocity of the joint after correction by the impedance model. Due to the decoupling of the dynamics of the two controllers through orthogonal projections, the controller adopts the error dynamics and stability analyses of the individual force and motion controllers within their respective subspaces. Consequently, the stability of the hybrid controller in closed-loop operation can be established.

5. Object Manipulation Control

In this paper, we present a segmented hybrid impedance motion/force control method designed to achieve the primary objective of wrapping around an obstacle and establishing multiple points of contact. By creating these multiple contact points, we aim to securely grasp a free-floating obstacle in space with enhanced robustness. This approach not only facilitates effective manipulation of objects but also improves stability during the interaction process. Furthermore, the robotic arm will not be confined to its end-effector or object geometries.

The object’s attitude control is achieved through a passive mechanical constraint strategy rather than active force control. Specifically, the force applied constantly by the end-effector and the configuration of the HRSM around the object create a geometrically constrained system that inherently keeps the object’s orientation to a minimum.

In the following section, we present a detailed analysis of object position control during space manipulation. Before proceeding with the control formulation, it is essential to establish the concept of the virtual end-effector. Unlike traditional robotic manipulation, where a single end-effector interacts with the object, our snake-like configuration creates multiple interaction points with the target object. We define the virtual end-effector as the collective set of links and joints that maintain contact with the obstacle during manipulation.

5.1. Object Position Control

The previous section detailed the manipulator’s position and force controller. The position PD controller is specifically designed to accurately achieve the desired positions; however, the manipulator is required to navigate around a non-cooperative object, which, for the sake of simplicity and to focus on controller development, is represented as a cylinder. The presence of this substantial mass introduces forces that the position controller alone cannot effectively manage, rendering it insufficient in this context. Consequently, there is a need for an additional control algorithm to complement the existing position controller, ensuring more robust handling of the dynamic interactions with the obstacle.

The trajectory planning needs to ensure that the obstacle will follow a preplanned trajectory. Given the assumption that the contact point between the closest link, called the virtual end-effector, and the obstacle does not change due to the applied force by the end effector, we can calculate a trajectory for our obstacle using a fifth-order polynomial function. A trajectory is calculated for the manipulator, which, of course, can be transformed to the obstacle trajectory by:

$${\mathrm{p}}_{\mathrm{O}}={\mathrm{T}}_{{\mathrm{C}}_{\mathrm{j}}}\left[\begin{array}{c}{\mathrm{p}}_{\mathrm{o}\mathrm{f}\mathrm{f}\mathrm{s}\mathrm{e}\mathrm{t}}+\mathrm{d}\\ 1\end{array}\right],$$

(37)

where ${\mathrm{p}}_{\mathrm{O}}$ is the position vector of the obstacle’s center in the global frame, ${\mathrm{T}}_{{\mathrm{C}}_{\mathrm{j}}}$ is the transformation matrix corresponding to closest joint, ${\mathrm{p}}_{\mathrm{o}\mathrm{f}\mathrm{f}\mathrm{s}\mathrm{e}\mathrm{t}}$ is the position vector of the contact point and $\mathrm{d}$ is the fixed offset vector from the contact point to the obstacle center.

Let ${\mathrm{T}}_{\mathrm{b},\mathrm{i}}=({\mathrm{p}}_{\mathrm{b},\mathrm{i}},{\mathrm{R}}_{\mathrm{b},\mathrm{i}})\in \mathrm{S}\mathrm{E}(3)$ be the initial pose of the manipulator, with ${\mathrm{p}}_{\mathrm{b},\mathrm{i}}$, ${\mathrm{R}}_{\mathrm{b},\mathrm{i}}$ being the initial position and orientation, respectively, of ${\mathrm{C}}_{\mathrm{j}}$. Let ${\mathrm{T}}_{\mathrm{b},\mathrm{f}}=\left({\mathrm{p}}_{\mathrm{b},\mathrm{f}},{\mathrm{R}}_{\mathrm{b},\mathrm{f}}\right)\in \mathrm{S}\mathrm{E}\left(3\right),$ with ${\mathrm{p}}_{\mathrm{b},\mathrm{f}}$, ${\mathrm{R}}_{\mathrm{b},\mathrm{f}}$ being the final position and orientation, respectively, of ${\mathrm{C}}_{\mathrm{j}}$. The desired E-E trajectory in each instant of time t ∈ [t_i, t_f] is given by:

$$0{\mathrm{p}}_{\mathrm{b},\mathrm{d}}(\mathrm{t})={\mathrm{p}}_{\mathrm{b},\mathrm{i}}+{\displaystyle \frac{\mathrm{s}\left(\mathrm{t}\right)}{‖{\mathrm{p}}_{\mathrm{b},\mathrm{f}}-{\mathrm{p}}_{\mathrm{b},\mathrm{i}}‖}}({\mathrm{p}}_{\mathrm{b},\mathrm{f}}-{\mathrm{p}}_{\mathrm{b},\mathrm{i}})$$

(38)

$$\dot{\mathrm{q}}(\mathrm{t})=\left[\begin{array}{c}{\dot{\mathrm{p}}}_{\mathrm{b},\mathrm{d}}\\ {\dot{\omega }}_{\mathrm{b},\mathrm{d}}\end{array}\right]\mathrm{M}{\mathrm{J}}^{\mathrm{T}}{\left(\mathrm{J}\mathrm{M}{\mathrm{J}}^{\mathrm{T}}\right)}^{†},$$

(39)

With $\mathrm{M}{\mathrm{J}}^{\mathrm{T}}{\left(\mathrm{J}\mathrm{M}{\mathrm{J}}^{\mathrm{T}}\right)}^{†}$ representing the dynamically consistent pseudoinverse Jacobian, and ${\dot{\mathrm{p}}}_{\mathrm{b},\mathrm{d}}$, ${\dot{\omega }}_{\mathrm{b},\mathrm{d}}$ representing the desired velocity and angular velocity (twist) of ${\mathrm{C}}_{\mathrm{j}}$.

The algorithm for obstacle position control is based on the principles of artificial potential fields. This approach employs both repulsive and attractive forces to facilitate collision-free trajectories. In our implementation, we generate forces that actively push and pull the manipulator toward the desired position. The newly defined attractive and repulsive forces are designed to counteract the forces generated by the obstacle, ensuring effective navigation in dynamic environments. These forces are translated to the manipulator as stabilization torque.

We propose a directional decomposition approach where the stabilizing forces are computed separately for each spatial direction. The stabilization force for each dimension is then computed using a proportional–exponential control approach, which provides a smooth transition between zero force and the necessary corrective force as the deviation increases. The potential force magnitude is defined as follows:

$${\mathrm{h}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{x}}}={\mathrm{k}}_{{\mathrm{p}1}_{\mathrm{x}}}({\mathrm{e}}^{-{\mathrm{k}}_{{\mathrm{p}2}_{\mathrm{x}}}{\mathrm{d}}_{\mathrm{x}}}-1),$$

(40)

$${\mathrm{h}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{y}}}={\mathrm{k}}_{{\mathrm{p}1}_{\mathrm{y}}}({\mathrm{e}}^{-{\mathrm{k}}_{{\mathrm{p}2}_{\mathrm{y}}}{\mathrm{d}}_{\mathrm{y}}}-1),$$

(41)

$${\mathrm{h}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{z}}}={\mathrm{k}}_{{\mathrm{p}1}_{\mathrm{z}}}({\mathrm{e}}^{-{\mathrm{k}}_{{\mathrm{p}2}_{\mathrm{z}}}{\mathrm{d}}_{\mathrm{z}}}-1),$$

(42)

where ${\mathrm{k}}_{{\mathrm{p}1}_{\mathrm{x}}}$, ${\mathrm{k}}_{{\mathrm{p}2}_{\mathrm{x}}}$, ${\mathrm{k}}_{{\mathrm{p}1}_{\mathrm{y}}}$, ${\mathrm{k}}_{{\mathrm{p}2}_{\mathrm{y}}}$, ${\mathrm{k}}_{{\mathrm{p}1}_{\mathrm{z}}}$ and ${\mathrm{k}}_{{\mathrm{p}2}_{\mathrm{z}}}$ are control gains that regulate the magnitude and responsiveness of the stabilization force, and ddd represents the distance between the desired and actual positions in each spatial dimension (X, Y, and Z). The control strategy also includes a spring–mass–damping model for the forces applied in each spatial dimension. This approach simulates spring-like behavior to stabilize the system around the desired position while dampening the oscillations through a damping coefficient. For each dimension, the control system computes a force based on the positional deviation and velocity difference, using both proportional control and spring-damping mechanics. Equations (40)–(42) then become as follows:

$${\mathrm{h}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{x}}}={\mathrm{k}}_{{\mathrm{p}1}_{\mathrm{x}}}({\mathrm{e}}^{-{\mathrm{k}}_{{\mathrm{p}2}_{\mathrm{x}}}{\mathrm{d}}_{\mathrm{x}}}-1)+{\mathrm{k}}_{\mathrm{s}}.{\mathrm{d}}_{\mathrm{x}}+{\mathrm{c}}_{\mathrm{d}}.{\mathrm{v}}_{\mathrm{x}},$$

(43)

$${\mathrm{h}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{y}}}={\mathrm{k}}_{{\mathrm{p}1}_{\mathrm{y}}}({\mathrm{e}}^{-{\mathrm{k}}_{{\mathrm{p}2}_{\mathrm{y}}}{\mathrm{d}}_{\mathrm{y}}}-1)+{\mathrm{k}}_{\mathrm{s}}.{\mathrm{d}}_{\mathrm{y}}+{\mathrm{c}}_{\mathrm{d}}.{\mathrm{v}}_{\mathrm{y}},$$

(44)

$${\mathrm{h}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{z}}}={\mathrm{k}}_{{\mathrm{p}1}_{\mathrm{z}}}({\mathrm{e}}^{-{\mathrm{k}}_{{\mathrm{p}2}_{\mathrm{z}}}{\mathrm{d}}_{\mathrm{z}}}-1)+{\mathrm{k}}_{\mathrm{s}}.{\mathrm{d}}_{\mathrm{z}}+{\mathrm{c}}_{\mathrm{d}}.{\mathrm{v}}_{\mathrm{z}},$$

(45)

where ${\mathrm{v}}_{\mathrm{x}}$, ${\mathrm{v}}_{\mathrm{y}}$ and ${\mathrm{v}}_{\mathrm{z}}$ are velocity errors in the X, Y and Z directions. ${\mathrm{k}}_{\mathrm{s}}$ the stiffness of the spring and ${\mathrm{c}}_{\mathrm{d}}$ is the damping coefficient.

The forces vectors can be computed as follows:

$${\mathrm{f}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{x}}}={\displaystyle \frac{{\mathrm{h}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{x}}}}{{\mathrm{d}}_{\mathrm{x}}}}\left({\mathrm{p}}_{{\mathrm{b}}_{\mathrm{x}}}-{\mathrm{p}}_{{\mathrm{b},\mathrm{i}}_{\mathrm{x}}}\right),$$

(46)

$${\mathrm{f}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{y}}}={\displaystyle \frac{{\mathrm{h}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{y}}}}{{\mathrm{d}}_{\mathrm{y}}}}\left({\mathrm{p}}_{{\mathrm{b}}_{\mathrm{y}}}-{\mathrm{p}}_{{\mathrm{b},\mathrm{i}}_{\mathrm{y}}}\right),$$

(47)

$${\mathrm{f}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{z}}}={\displaystyle \frac{{\mathrm{h}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{z}}}}{{\mathrm{d}}_{\mathrm{z}}}}\left({\mathrm{p}}_{{\mathrm{b}}_{\mathrm{z}}}-{\mathrm{p}}_{{\mathrm{b},\mathrm{i}}_{\mathrm{z}}}\right),$$

(48)

The torque exerted by the force is given by:

$${\mathsf{\tau }}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{x}}}=\mathrm{M}\left(\mathrm{q}\right){\mathrm{J}}_{\mathrm{i}}^{\mathrm{T}}(\mathrm{q}){\mathrm{f}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{x}}},$$

(49)

$${\mathsf{\tau }}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{y}}}=\mathrm{M}\left(\mathrm{q}\right){\mathrm{J}}_{\mathrm{i}}^{\mathrm{T}}(\mathrm{q}){\mathrm{f}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{y}}},$$

(50)

$${\mathsf{\tau }}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{z}}}=\mathrm{M}\left(\mathrm{q}\right){\mathrm{J}}_{\mathrm{i}}^{\mathrm{T}}(\mathrm{q}){\mathrm{f}}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{z}}},$$

(51)

where ${\mathrm{J}}_{\mathrm{i}}^{\mathrm{T}}\left(\mathrm{q}\right)$ is the transpose of the Jacobian matrix forming the first joint to ${\mathrm{C}}_{\mathrm{j}}$.

To maintain system stability, the control strategy uses projection matrices to decompose the applied torques into distinct subspaces. The torques along the X- and Y-axes are projected into their respective subspaces to manage the directional stability, while the torque along the Z-axis is handled separately in a dedicated subspace. This approach ensures precise control of each spatial component, reducing the cross-axis interference and enhancing the overall system robustness. The projection matrices can be written as follows:

$$P_x=I-{\left(A_{x}J\right)}^T{\left(A_{x}J{M}^{-1}{\left(A_{x}J\right)}^T\right)}^{-1}{A}_{x}J{M}^{-1},$$

(52)

$$P_y=I-{\left(A_{y}J\right)}^T{\left(A_{y}J{M}^{-1}{\left(A_{y}J\right)}^T\right)}^{-1}{A}_{y}J{M}^{-1},$$

(53)

$$P_z=I-{\left(A_{z}J\right)}^T{\left(A_{z}J{M}^{-1}{\left(A_{z}J\right)}^T\right)}^{-1}{A}_{z}J{M}^{-1},$$

(54)

where $A_x$, $A_y$ and $A_z$ are Pfaffian constraints matrices in the joint space for the X, Y and Z components.

The final combined hybrid torque can be represented by:

$${\mathsf{\tau }}_{\mathrm{H}\mathrm{y}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{d}}=P_x{\mathsf{\tau }}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{x}}}+{P_y\mathsf{\tau }}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{y}}}+\left({I-P}_z\right){\mathsf{\tau }}_{{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{\mathrm{z}}},$$

(55)

5.2. Object Collision Avoidance

In the operation of hyper-redundant manipulators, avoiding collisions with surrounding obstacles is critical to ensure safe and efficient execution of tasks. The obstacle avoidance torque is designed to generate forces that push the manipulator away from detected obstacles, preventing contact while preserving the planned trajectory as much as possible. The collision detection and avoidance system continuously monitors the distances between the manipulator’s joints and nearby obstacles. When the manipulator approaches an obstacle, a repulsive force is generated to guide it away, with the torque being applied in the direction of the closest joint to the obstacle. The avoidance strategy consists of several steps, as detailed below.

For each joint, the Euclidean distance to the nearest obstacle is calculated. To enhance the collision detection resolution and monitor the whole length of the object, the positions of each joint are projected along the Z-axis in both directions (above and below the joint). This creates two reference points to account for the obstacle height. The repulsive force is designed using an exponential relationship to ensure that the closer the manipulator gets to an obstacle, the stronger the force pushing it away becomes. The potential force magnitude due to the obstacle and the force vector are defined as follows:

$${\mathrm{h}}_{\mathrm{a}\mathrm{v}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}}={\mathrm{k}}_{\mathrm{p}1}({\mathrm{e}}^{{\mathrm{k}}_{\mathrm{p}2}\left({\mathrm{d}}_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}\mathrm{t}\mathrm{y}}+{\mathrm{d}}_0-{\mathrm{d}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}-1),$$

(56)

$${\mathrm{f}}_{\mathrm{a}\mathrm{v}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}}={\displaystyle \frac{{\mathrm{h}}_{\mathrm{a}\mathrm{v}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}}}{{\mathrm{d}}_{\mathrm{m}\mathrm{i}\mathrm{n}}}}\left({\mathrm{p}}_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{s}\mathrm{t}}-{\mathrm{p}}_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{d}}\right),$$

(57)

${\mathrm{d}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ is the minimum distance between the projected points and the obstacle, ${\mathrm{d}}_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}\mathrm{t}\mathrm{y}}+{\mathrm{d}}_0$ defines the distance at which the repulsive force becomes active, ${\mathrm{p}}_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{s}\mathrm{t}}$ is the closest joint to the obstacle position and ${\mathrm{p}}_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{d}}$ is the desired position on the trajectory. The torque exerted by the force is given by:

$${\mathsf{\tau }}_{\mathrm{a}\mathrm{v}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}}=\mathrm{M}\left(\mathrm{q}\right){\mathrm{J}}^{\mathrm{T}}\left(\mathrm{q}\right){\mathrm{f}}_{\mathrm{a}\mathrm{v}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}},$$

(58)

Since the manipulator wraps around the obstacle, any movement of the manipulator joints to avoid contact with external objects automatically guides the obstacle along the intended path.

5.3. Object Self-Collision Avoidance

During operation, self-collision scenarios are likely to occur, especially in winding configurations. Self-collision is particularly common between the virtual end-effector (last 10 links) and the rest of the manipulator. To overcome this problem, a self-collision avoidance algorithm has been implemented.

The algorithm calculates the distances between all the joints. If the distance drops below a certain threshold, a repulsive force is applied to both joints in opposite directions. The potential force magnitude is defined as follows:

$${\mathrm{h}}_{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}}={\mathrm{k}}_1\left({\displaystyle \frac{1}{1+{\mathrm{e}}^{{\mathrm{k}}_2\left({\mathrm{d}}_{\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}-}{\mathrm{d}}_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}\mathrm{t}\mathrm{y}}+{\mathrm{d}}_0\right)}}}\right),$$

(59)

where ${\mathrm{k}}_1$ is the proportional gain for the force calculation, ${\mathrm{k}}_2$ is the exponential gain for the force calculation, ${\mathrm{d}}_{\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}}$ is the distance between the closest joints and ${\mathrm{d}}_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}\mathrm{t}\mathrm{y}}+{\mathrm{d}}_0$ is the safety distance threshold.

The force vector can be computed as follows:

$${\mathrm{f}}_{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f} \mathrm{a}\mathrm{v}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}}=\left[\begin{array}{c}{0}^{3\times 1}\\ {\displaystyle \frac{{\mathrm{h}}_{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}}}{{\mathrm{d}}_{\mathrm{m}\mathrm{i}\mathrm{n} \mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}}}}\left({\mathrm{p}}_{\mathrm{i}}-{\mathrm{p}}_{\mathrm{j}}\right)\end{array}\right],$$

(60)

where ${\mathrm{p}}_{\mathrm{i}}$ and ${\mathrm{p}}_{\mathrm{j}}$ represent the two closest joints. The task of self-avoidance, although always active, is considered a secondary task. In this context the augmented Jacobian weighted pseudo inverse is used. The torque exerted is given by:

$${\mathsf{\tau }}_{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}\text{-}\mathrm{a}\mathrm{v}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}}=\left(\mathrm{I}-{\mathrm{J}}_{\mathrm{a}\mathrm{u}\mathrm{g}}^{\#}(\mathrm{t}){\mathrm{J}}_{\mathrm{a}\mathrm{u}\mathrm{g}}(\mathrm{t})\right){\mathrm{J}}_{\mathrm{i}}^{\mathrm{T}}(\mathrm{t}){\mathrm{f}}_{\mathrm{a}\mathrm{v}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}},$$

(61)

5.4. Final Control Input

The final controller in this paper is the base controller. The base attitude is controller can be expressed as follows:

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}\mathrm{H}=\mathrm{U}-\omega \times \mathrm{H},$$

(62)

where H is the angular momentum vector, U is the control torque input calculated by the controller and $\omega$ is the angular velocity vector of the base. The control input U can be expressed as follows:

$$\mathrm{U}={\mathrm{K}}_{\mathrm{p}}{\mathrm{e}}_{\mathrm{A}}+{\mathrm{K}}_{\mathrm{d}}{\mathrm{e}}_{\dot{\mathrm{A}}},$$

(63)

where ${\mathrm{e}}_{\mathrm{A}}$ and ${\mathrm{e}}_{\dot{\mathrm{A}}}$ are the attitude error and rate of change error, respectively. ${\mathrm{K}}_{\mathrm{p}}$ and ${\mathrm{K}}_{\mathrm{d}}$ are the proportional and derivative gain constants.

The final controller output designed can be summarized in the following equation:

$${\mathsf{\tau }}_{\mathrm{a}\mathrm{l}\mathrm{l}}=\mathrm{U}{\mathrm{M}}_{\mathrm{b}}+\mathrm{P}\left(\mathrm{q}\right)(\mathrm{M}\left(\mathrm{q}\right)({\ddot{\mathrm{q}}}_{\mathrm{H}}\left(\mathrm{t}\right)+{\mathrm{K}}_{\mathrm{p}1}\left({\mathrm{q}}_{\mathrm{d}}-\mathrm{q}\right)+{\mathrm{K}}_{\mathrm{d}1}\left({\dot{\mathrm{q}}}_{\mathrm{H}}-\dot{\mathrm{q}}\right)))+\left(\mathrm{I}-\mathrm{P}\left(\mathrm{q}\right)\right)({\mathrm{J}}^{\mathrm{T}}(\mathrm{q})({\mathrm{F}}_{\mathrm{d}}+{\mathrm{K}}_{\mathrm{f}\mathrm{p}}{\mathrm{F}}_{\mathrm{e}}+{\mathrm{K}}_{\mathrm{f}\mathrm{i}}\int {\mathrm{F}}_{\mathrm{e}}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}-{\mathrm{K}}_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{p}}\nu ))+{\mathsf{\tau }}_{\mathrm{H}\mathrm{y}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{d}}+{\mathsf{\tau }}_{\mathrm{a}\mathrm{v}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}}+{\mathsf{\tau }}_{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}\text{-}\mathrm{a}\mathrm{v}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}},$$

(64)

with ${\mathrm{M}}_{\mathrm{b}}$ representing the base mass.

6. Simulation Analysis

The segmented hybrid impedance control method, in conjunction with the hyper-redundant robotic arm system’s dynamic model provided, collision detection algorithms, and contact collision force models, is used to establish the dynamic model of the HRSM during end-effector operation and perform simulation analysis.

In this section, we present two simulated application scenarios for the HRSM. We apply the control strategy outlined in Section 4 to the theoretical dynamics model described in Section 3. Each of the 24 links in the designed multi-rigid-body HRSM possesses identical properties. For reference,

Table 1. Rotation axis of the joints in scenario one.

Number	Rotation Axis	Body Number	Rotation Axis	Body Number	Rotation Axis
1	[1 0 0]T	9	[0 0 1]T	17	[1 0 0]T
2	[0 0 1]T	10	[0 0 1]T	18	[0 0 1]T
3	[1 0 0]T	11	[0 0 1]T	19	[0 1 0]T
4	[0 0 1]T	12	[0 0 1]T	20	[0 0 1]T
5	[1 0 0]T	13	[0 0 1]T	21	[1 0 0]T
6	[0 0 1]T	14	[0 0 1]T	22	[0 0 1]T
7	[0 0 1]T	15	[1 0 0]T	23	[1 0 0]T
8	[0 0 1]T	16	[0 0 1]T	24	[0 0 1]T

provides the rotation axis direction for the manipulator joints for the first scenario.

Table 2. Link geometry and mass parameters.

Body Number	Mass (kg)	First Inertia (kg·m)	Moment of Inertia (kg·m2)	Installation Location (m)
i	1.95	[0 0.315 0]T	diag(0.0582 0.0027 0.056)	[0 0.2 0]T

shows the material parameters of the contact model. The surface material of the robotic arm is hard rubber, with a low Young’s modulus and a high Poisson’s ratio, and due to the strong elasticity of the material, the recovery coefficient is also high. The target surface material is hard aluminum alloy.

Table 3. Contact model parameters.

Physical Quantity	Value	Physical Quantity	Value
$\mathrm{Robot} \mathrm{surface} \mathrm{elastic} \mathrm{modulus} {(\mathrm{E}}_1/\mathrm{M}\mathrm{P}\mathrm{a})$	7.84	$\mathrm{Target} \mathrm{surface} \mathrm{elastic} \mathrm{modulus} ({\mathrm{E}}_2/\mathrm{G}\mathrm{P}\mathrm{a})$	70
$\mathrm{Robot} \mathrm{surface} \mathrm{Poisson}’\mathrm{s} \mathrm{ratio} \left({\nu }_1\right)$	0.47	$\mathrm{Target} \mathrm{surface} \mathrm{Poisson}’\mathrm{s} \mathrm{ratio} \left({\nu }_2\right)$	0.3
$\mathrm{Force} \mathrm{coefficient} \left(\mathrm{n}\right)$	1.5	$\mathrm{Recovery} \mathrm{coefficient} \left({\mathrm{C}}_{\mathrm{r}}\right)$	0.7
$\mathrm{Dynamic} \mathrm{friction} \mathrm{coefficient} \left({\mathsf{\mu }}_{\mathrm{d}}\right)$	0.2	$\mathrm{Static} \mathrm{friction} \mathrm{coefficient} \left({\mathsf{\mu }}_{\mathrm{s}}\right)$	0.4
$\mathrm{Stribeck} \mathrm{model} \mathrm{index} ({\mathsf{\delta }}_{\mathrm{s}}$)	2	$\mathrm{Stribeck} \mathrm{speed} ({\widehat{\mathrm{v}}}_{\mathrm{s}}/\left(\mathrm{m}/\mathrm{s}\right)$)	0.1
$\mathrm{Robot} \mathrm{surface} \mathrm{elastic} \mathrm{modulus} {(\mathrm{E}}_1/\mathrm{M}\mathrm{P}\mathrm{a})$	7.84	$\mathrm{Target} \mathrm{surface} \mathrm{elastic} \mathrm{modulus} ({\mathrm{E}}_2/\mathrm{G}\mathrm{P}\mathrm{a})$	70
$\mathrm{Robot} \mathrm{surface} \mathrm{Poisson}’\mathrm{s} \mathrm{ratio} \left({\nu }_1\right)$	0.47	$\mathrm{Target} \mathrm{surface} \mathrm{Poisson}’\mathrm{s} \mathrm{ratio} \left({\nu }_2\right)$	0.3

presents the inertia parameters and geometric data for the manipulator links, indexed from i = 1 to 24. and

Table 4. Rotation axis of the joints in scenario two.

Number	Rotation Axis	Body Number	Rotation Axis	Body Number	Rotation Axis
1	[0 1 0]T	9	[1 0 0]T	17	[0 0 1]T
2	[0 0 1]T	10	[0 0 1]T	18	[0 0 1]T
3	[1 0 0]T	11	[1 0 0]T	19	[0 0 1]T
4	[0 0 1]T	12	[0 0 1]T	20	[0 0 1]T
5	[1 0 0]T	13	[1 0 0]T	21	[0 0 1]T
6	[0 0 1]T	14	[0 0 1]T	22	[0 0 1]T
7	[1 0 0]T	15	[0 0 1]T	23	[0 0 1]T
8	[0 0 1]T	16	[0 0 1]T	24	[0 0 1]T

provides the rotation axis direction for the manipulator joints for the second scenario.

6.1. First Scenario

This scenario sketched in Figure 6 and depicted in Figure 7 is created to validate the new segmented hyper-impedance controller, focusing mainly on the impedance behavior of the manipulator with both the non-cooperative object and the base treated as fixed rigid bodies and the manipulator segmented into two segments. This case can be compared in the real world to a manipulator wrapping around a truss structure to achieve a better end-effector position and velocity error. In the second scenario, our experimental set-up mirrors the challenges encountered in orbital debris removal missions. The diverse nature of space debris, varying in geometry, mass distribution, and material composition, presents a complex manipulation challenge that validates the adaptability of our control approach. The controller’s ability to maintain stable impedance characteristics while handling objects demonstrates its potential application in space debris removal operations. This flexibility is particularly valuable as debris can range from defunct satellites with regular geometries to fragmented components with unpredictable mass distributions.

The force controller parameters are as follows:

$${\mathrm{K}}_{\mathrm{p}\_\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{e}}=15,{\mathrm{K}}_{\mathrm{i}\_\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{e}}=7$$

$${\mathrm{F}}_{\mathrm{d}}=10 \mathrm{N}$$

The hybrid position force controller parameters are as follows:

$${\mathrm{K}}_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{p}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\right[10 10 10 10 10 10\left]\right)$$

$${\mathrm{K}}_{\mathrm{p}\_\mathrm{s}\mathrm{e}\mathrm{c}1}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\right[10 10 \cdots 10 10\left]\right),{\mathrm{K}}_{\mathrm{d}\_\mathrm{s}\mathrm{e}\mathrm{c}1}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\right[16 16 \cdots 16 16\left]\right)$$

The impedance control model parameters for 10 joints (joint 6 to joint 15) are as follows:

$${\mathrm{M}}_{\mathrm{H},\mathrm{d}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(100\cdots 100),{\mathrm{B}}_{\mathrm{H},\mathrm{d}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(2000\cdots 2000)$$

Since there will be multiple collisions between the robotic arm and the contact target in a multi-point contact scenario, the motion control error of tracking the trajectory of the first segment before the second segment of the robotic arm achieves stable leaning is large, so the winding process needs to be performed at different time periods.

In the numerical simulation, the initial position of the manipulator is around the target, but it has not yet made contact with the target. The first stage of the motion process is 0–10 s. The 6–15 joints of the robotic arm converge toward the target according to the designed expected trajectory. After contact and collision with the target, a force/position hybrid impedance controller is used instead to make the 14th joint track the expected value with a normal contact force of 10 N; the second stage is 10–20 s. Based on the premise that the first segment of the robotic arm achieves stable contact, the second segment performs position control according to the designed desired trajectory shown in Figure 8a,b to complete the desired robotic arm manipulation. We introduce noise to the link 14 normal force sensor, as shown if Figure 9.

The impedance control is only applied to joints 6 to 15 because they are the section in contact of the first segment of the robotic arm because this can minimize the impact of the first segment’s motion on the second segment while ensuring that the first segment of the robotic arm exhibits good impedance characteristics. The simulation results are compared with the situation where after the first segment of the manipulator comes into contact with an external object, no resistance is applied and only force/position hybrid control is used.

The impedance model adjusts the manipulator’s impedance characteristics, leading to a noticeable reduction in the number of contacts between the manipulator and the object, as shown in Figure 10a,b. This outcome aligns with the design objectives of the impedance controller, addressing the limitations observed in previous work where impedance was not implemented. Fewer contacts result in more robust control of the HRSM, offering better protection from physical damage and ensuring stable multi-point contact.

In contrast, Figure 11a,b present the contact collision force response without low-frequency filtering. When only the force/position hybrid control method is employed, the robotic arm experiences multiple intense collisions with external objects, with the instantaneous contact forces exceeding 650 N. These high forces can cause significant impacts on the robot’s joints, potentially leading to instability and compromising the HRSM hardware.

Impedance control effectively imparts “softer” characteristics to the robotic arm, aligning with the original design goal of achieving active compliance control. This control strategy allows the manipulator to quickly adapt to unpredictable contact scenarios, particularly when anchoring and applying forces to external objects. As demonstrated in Figure 12a,b, the manipulator successfully applies a desired force of 10 N on the obstacle, exhibiting clear impedance behavior. In contrast, when impedance control is disabled, the manipulator fails to exhibit this level of compliance, highlighting the crucial role of impedance control in achieving the desired dynamic interaction with the environment. According to Figure 13a,b and Figure 14a,b the end-effector exhibit better position and velocity under impedance control influence.

6.2. Second Scenario

The simulated scenario is sketched in Figure 15 and illustrated in Figure 16, where the manipulator wraps around an obstacle with a configuration stated in Table 4, grasps it, and manipulates it along a planned trajectory while avoiding additional obstacles. Initially, the manipulator is positioned near the target but has not yet established contact. In this case, the manipulator is treated as a single segment (joint 1 to joint 24) and the desired force is applied by the end-effector. The designed desired trajectory shown in Figure 17a,b describe the desired robotic arm manipulation.

Figure 18a,b shows the number of contacts between the manipulator and the object where we can notice a significant reduction of contact separation. During the first simulation phase, the motion planning is coupled with a force controller to ensure secure object grasping. The end-effector applies a consistent force of 5 N to maintain a stable grip, as shown in Figure 19a,b and Figure 20a,b. Impedance control is employed by the virtual end-effector (joint 15 to joint 24) to ensure compliance during the contact and mitigate forces exerted by the obstacle. After securing the object, the manipulator initiates its movement.

The Euclidean position error of the object (Figure 21a,b) remains within permissible bounds, given the manipulator’s width and the obstacle’s size. We introduce noise to the end-effector contact sensor, as shown if Figure 9. The system stability is evaluated by verifying that the sum of the external forces and torques remains within acceptable limits, as demonstrated in Figure 22a,b and Figure 23a,b. The base’s Euler angles also exhibit a negligible influence on the system efficiency (Figure 24a,b).

Finally, base attitude control plays a critical role in counteracting destabilizing external forces. This integration of trajectory tracking, force control, and stability assessment highlights the system’s capability to manage complex dynamic tasks effectively.

The force controller parameters are as follows:

$${\mathrm{K}}_{\mathrm{p}\_\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{e}}=15,{\mathrm{K}}_{\mathrm{i}\_\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{e}}=7$$

$${\mathrm{F}}_{\mathrm{d}}=5 \mathrm{N} (1.5–10 \mathrm{s}); {\mathrm{F}}_{\mathrm{d}}=10 \mathrm{N} (10–30 \mathrm{s}); \mathrm{Object} \mathrm{mass}=100 \mathrm{kg}$$

The hybrid position force controller parameters are as follows:

$${\mathrm{K}}_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{p}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\right[10 10 10 10 10 10\left]\right)$$

$${\mathrm{K}}_{\mathrm{p}\_\mathrm{s}\mathrm{e}\mathrm{c}1}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\right[10 10 \cdots 10 10\left]\right),{\mathrm{K}}_{\mathrm{d}\_\mathrm{s}\mathrm{e}\mathrm{c}1}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\right[16 16 \cdots 16 16\left]\right)$$

The impedance control model parameters for 11 joints (joint 14 to joint 24) are as follows:

$${\mathrm{M}}_{\mathrm{H},\mathrm{d}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(400\cdots 400),{\mathrm{B}}_{\mathrm{H},\mathrm{d}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(5000\cdots 5000)$$

7. Conclusions

In this paper, we addressed the challenge of compliance with multiple collision points in hyper-redundant space manipulators. We developed a hybrid impedance controller algorithm specifically designed to enable robust interaction and collision handling with obstacles. Additionally, our approach integrated object stabilization and both object and self-collision avoidance mechanisms, facilitating the manipulation of large objects in complex environments.

Comparative analysis of the model both with and without impedance characteristics demonstrated the robustness of the control theory against the uncertainties associated with multi-point collisions. The practical implementation of this method further validated its effectiveness, highlighting the advantages of incorporating impedance control in enhancing the performance and reliability of hyper-redundant manipulators in dynamic settings. This work can be further enhanced by integrating machine learning techniques to dynamically adapt the impedance parameters based on the real-time conditions, enabling more precise and responsive control. Additionally, incorporating methods specifically designed to address residual vibrations, in conjunction with the current impedance-based approach, could significantly improve the overall performance and stability of the control architecture. These advancements have the potential to enhance the system robustness.