3.2.2. Feasibility Analysis of Grasping

The gripper is rotated synchronously with the fruit to separate the fruits from the stalk. During this period, the fingers contact the target fruit, and the two objects do not allow to slide against each other to achieve stable grasping. Thus, the whole system [27,28] is in equilibrium. The following relationships of mechanical and dynamic need to be satisfied:

$$\begin{cases} Gf = \omega \\ J^T f = \pi \end{cases} \tag{10}$$

$$\begin{cases} G^T \mu = \dot{\mathbf{x}} \\ J\dot{q} = \dot{\mathbf{x}} \end{cases} \tag{11}$$

where *ω* is the external force vector; *τ* is the joint torque vector; *f* is the total contact vector acting on the fruit; *<sup>u</sup>* is the velocity vector; . *<sup>q</sup>* is the joint velocity vector; . *x* is the contact velocity vector; *J* is the gripper Jacobian matrix and determinant; and *G* is the gripping matrix of the gripper.

Therefore, if the object is required to be constrained entirely, then the relative acceleration vector in the contact force space must be zero. The finger ends and the target fruit are relatively static. For a given *τ*, the above-mentioned feasible velocity must be zero in all directions of binding force action. For this reason, for a given system state and external force, the amplitude *λ* of normal contact force should be greater than 0 (*λ<sup>C</sup>* > 0) to satisfy the linear complementary condition of contact constraint. *λ<sup>C</sup>* includes normal contact force *FN* and friction force *Ff* . It must satisfy the constraint condition *Ff* ≤ *μFN*. (Figure 7).

**Figure 7.** The feasibility analysis of gripper gripping. Note: arrows and dashed lines say finger designs are inspired by hand grips.

So, at the contact point *Ci*, the contact force *FCi* <sup>=</sup> *FCi*,*<sup>n</sup> FCi*,*<sup>t</sup> <sup>T</sup>* satisfies the constraint condition:

$$F\_{\mathbb{C}\_i, t} = \sqrt{\lambda\_{i, t\_1}^2 + \lambda\_{i, t\_2}^2} \le \mu F\_{\mathbb{C}\_i, n} \tag{12}$$

where *λi*,*t*<sup>1</sup> and *λi*,*t*<sup>2</sup> are special solution vectors of *λC*. They correspond to the magnitude and curl of the friction at *Ci*. *λ<sup>i</sup>* is the amplitude component of the normal contact force at *Ci*. Let *fi* × *λi*= *FCi*,*n*, and the constant factor *fi* ≥ *λ*2 *i*,*t*<sup>1</sup> +*λ*<sup>2</sup> *i*,*t*<sup>2</sup> /*μλ<sup>i</sup>* . For *λ* = *λC*+*aλ*0, all normal contact force amplitudes *λ<sup>i</sup>* > 0 were guaranteed. The dynamic model of grasping constraint can be satisfied by the feasible solution of contact force. *a* is a scalar quantity.
