*3.1. Dynamic Model of the System*

The bodies of the mechanism are assumed to be rectangular beams whose lengths are *L*<sup>1</sup> = 2 m, *L*<sup>2</sup> = 8 m, and *L*<sup>3</sup> = 5 m and whose masses are *m*<sup>1</sup> = 100 kg, *m*<sup>2</sup> = 400 kg, and *m*<sup>3</sup> = 250 kg, respectively. The position vector at point *D* is **r***<sup>D</sup>* = <sup>−</sup> *<sup>L</sup>*<sup>1</sup> <sup>2</sup> 0 0 *<sup>T</sup>* . The point *G* is located at the centre of mass of body 1. The double-step semi-recursive formulation described in Section 2.1.1 is used to model the four-bar mechanism.

The four-bar mechanism is actuated using the sinusoidal reference input signal, which is taken as *Uref* = 10 sin (0.4*πk*), where *k* is the simulation run time. The simulations are performed for 5 s. The hydraulic circuit consists of a double-acting hydraulic cylinder, connecting hoses 1 and 2, a 4/3 directional control valve, a pressure relief valve, a connecting hose of volume *Vp*, a differential pump of pressure *pp* , and a tank with a constant pressure source *pT*.

**Figure 2.** Hydraulically actuated four-bar mechanism. The mechanism is actuated by a differential pressure pump. **Cv***<sup>a</sup>* ,**Cv***<sup>b</sup>* ,**Cv***<sup>c</sup>* , and **Cv***<sup>d</sup>* represent the semi-empiric flow rate coefficients at the *a*, *b*, *c*, and *d* ports of the 4/3 directional control valve. Grey rectangles indicate the pressure sensors on the control volumes *Vp* and *V*1.

The lumped fluid theory described in Section 2.1.2 is used to compute the pressures within the hydraulic circuit. In the application of the lumped fluid theory, the hydraulic circuit can be divided into three control volumes *Vp*, *V*1, and *V*2. The pressure derivatives *p*˙ *<sup>p</sup>*, *p*˙1, and *p*˙2 through these volumes can be computed as

$$\begin{aligned} \dot{p}\_p &= \frac{k\_p + p\_p k\_0}{V\_p} (Q\_p - Q\_R - Q\_{d\_1})\\ \dot{p}\_1 &= \frac{k\_p + p\_1 k\_0}{V\_1} (Q\_{d\_1} - A\_1 \dot{s})\\ \dot{p}\_2 &= \frac{k\_p + p\_2 k\_0}{V\_2} (A\_2 \dot{s} - Q\_{d\_2}) \end{aligned} \tag{32}$$

where *Qd*<sup>1</sup> and *Qd*<sup>2</sup> are the flow rates in the control volumes 1 and 2. In Equation (32), *Qp* and *QR* are the pump flow rate and flow rate through the pressure relief valve, respectively. The flow rates *QR*, *Qd*<sup>1</sup> , and *Qd*<sup>2</sup> can be computed by employing Equations (9) and (10), respectively. The constant hydraulic parameters are tabulated in Table 1. In Equation (32), *s*˙ is the actuator velocity, which can be determined from the actuator position vector **s**. Following Figure 2, the vector **s** can be calculated from the position vectors **r***<sup>G</sup>* and **r***<sup>D</sup>* as

$$\begin{aligned} \mathbf{s} &= \mathbf{r}\_G - \mathbf{r}\_D\\ \dot{\mathbf{s}} &= \frac{d \mid \mathbf{s} \mid}{\Delta t} = \dot{\mathbf{s}} \cdot \frac{\mathbf{s}}{\lfloor \mathbf{s} \rfloor} = \dot{\mathbf{r}}\_G \cdot \frac{\mathbf{s}}{\lfloor \mathbf{s} \rfloor} \end{aligned} \tag{33}$$

where **r**˙*<sup>G</sup>* is the velocity vector of point *G*. The control volumes *V*<sup>1</sup> and *V*<sup>2</sup> appearing in Equation (32) can be calculated as follows:

$$\begin{aligned} V\_1 &= V\_{h\_1} + A\_1 l\_1\\ V\_2 &= V\_{h\_2} + A\_2 l\_2 \end{aligned} \tag{34}$$

where *Vh*<sup>1</sup> , *Vh*<sup>2</sup> , and *Vp* are the control volumes of the respective hoses, as described in Table 1. In Equation (34), *l*<sup>1</sup> and *l*<sup>2</sup> are the lengths of the piston side and the piston-rod side chambers, respectively. *l*<sup>1</sup> and *l*<sup>2</sup> can be calculated with the vector **s** as

$$\begin{aligned} l\_1 &= l\_{1\_0} - |\mathbf{s}| + s\_0 \\ l\_2 &= l\_{2\_0} + |\mathbf{s}| - s\_0 \end{aligned} \tag{35}$$

where *l*<sup>10</sup> and *l*<sup>20</sup> are the initial piston side length and the initial piston-rod side length, respectively. *l*10 and *l*20 are computed from the length of cylinder *l*, which is given in Table 1.

**Table 1.** Parameters of the hydraulic circuit.


Using the vector **s**, the hydraulic force **F***<sup>h</sup>* produced by the double-acting cylinder can be calculated as

$$\mathbf{F}\_{\rm h} = \begin{bmatrix} \frac{s\_{\rm X}}{|\mathbf{s}|} F\_{\rm h} & \frac{s\_{\rm Y}}{|\mathbf{s}|} F\_{\rm h} & \frac{s\_{\rm Z}}{|\mathbf{h}|} F\_{\rm h} \end{bmatrix}^{T} \tag{36}$$

where *Fh* is computed from Equation (13). The hydraulic force vector **F***<sup>h</sup>* is combined with the external force vector **f***<sup>i</sup>* to calculate **Q** in Equation (3). The resultant equations of motion (15) are formulated for the hydraulically driven four-bar mechanism. Equations (15) are solved by using an implicit single-step trapezoidal integration scheme in a monolithic approach, which was described in Section 2.1.3.

#### 3.1.1. Real and Estimation Models

In this study, three dynamic versions of the mechanism are used to demonstrate the implementation of the parameter estimation algorithm. One of the models is the real model. The sensor measurements **o** are taken from the real model. The modelling errors are introduced in the force model of the estimation model with respect to the real model. The properties of the estimation model and the simulation model are the same. In Table 2, the properties of the real model, the estimation model, and the simulation model are provided. Note that the simulation model is used in this study to demonstrate the differences between the simulated world and the real world.

**Table 2.** Properties of the real model, the estimation model, and the simulation model. Errors in the simulation model and the estimation model are given in comparison to the real model. *s*<sup>10</sup> , *pp*<sup>0</sup> , and *p*<sup>10</sup> represent the initial actuator position, the initial pump pressure, and the initial pressure on the piston side as the system states. The system parameters **Cv***<sup>a</sup>* , **Cv***<sup>b</sup>* , **Cv***<sup>c</sup>* , **Cv***<sup>d</sup>* , *k*0, and *kp* represent the semi-empiric flow rate coefficient at the *a*, *b*, *c*, and *d* ports of the directional control valve, the flow gain, and the pressure flow coefficients, respectively.


As in practise, the minimum and maximum points on the characteristic curves of a directional control valve can be determined from the manufacturer's catalogues. Using this limited information, the characteristic curves are defined linearly at all ports of the directional control valve in the cases of the estimation model and the simulation model. The linear characteristic curves are implemented by using the minimum and maximum values of the semi-empiric flow rate coefficients **Cv***<sup>a</sup>* , **Cv***<sup>b</sup>* , **Cv***<sup>c</sup>* , and **Cv***<sup>d</sup>* at the valve closing and the valve opening positions, respectively. The linear characteristic curves of the directional control valve affect the dynamics of the estimation model throughout the simulation runtime. In the case of the real model, the characteristic curves of the directional control valve are unclear and can be non-linear. With Equation (21), the non-linear characteristic curves of the directional control valve are implemented using **Cv***<sup>a</sup>* , **Cv***<sup>b</sup>* , **Cv***<sup>c</sup>* , and **Cv***<sup>d</sup>* in the hydraulic circuit of the real model. Similarly, the initial actuator positions *s*<sup>10</sup> of the real model and the estimation model are different. Note that the initial relative joint coordinates of the bodies in the system can be found from *s*<sup>10</sup> and *s*˙10 by using geometrical relationships. To avoid instabilities in the integration process, the simulations are started in the static equilibrium position, the details of the mechanism of which can be found in [34].

#### 3.1.2. Sensor Measurements

In this study, the measurable observations **o** = *s pp p*<sup>1</sup> *T* are taken from the real model. In the real model, the actuator position sensor measures the actuator position *s* [76]. Gauge pressure sensors are used for the pressure measurements *pp* and *p*<sup>1</sup> [34]. These pressure sensors measure the pressure with respect to the atmospheric pressure. The pressure sensors are installed on their respective volumes, as also shown in Figure 2. The numerical values of the standard deviation, as mentioned in Equation (29), are taken as *σ <sup>s</sup>* = 1.12 × <sup>10</sup>−<sup>3</sup> m and *<sup>σ</sup> <sup>p</sup>* = 1.5 × <sup>10</sup><sup>5</sup> Pa for the actuator and pressure sensors, respectively.
