Joint Position Bounds in Resolved-Acceleration Control: A Comparison

Abstract

The implementation of human-friendly robots is based on the deployment of robots that can safely and effectively work with humans in various environments. To this end, enforcing joint limits in planning and control play a fundamental role in avoiding the robot to exceed its physical constraint and preventing joint damages or failures that could lead to unpredictable behavior or compromised safety. However, the implementation of such limitations in instantaneous controllers is not trivial when position, velocity, and acceleration limits are all considered together. In this work, we compare three State-of-the-Art methods, namely the P-Step Ahead Predictor, the Control Barrier Function, and Invariance. Finally, we select the most performing one applied in a real use case based on a UR5e manipulator for a picking task where hitting joint limits may represent an issue.

The Authors would like to acknowledge Dr. Matteo Parigi Polverini for the implementation of the joint position constraint based on invariance and Dr. Niels Dehio for his valuable help in reviewing the paper.

Appendix

In [4], the invariance or viability control approach is implemented, extending the method to accommodate larger sampling times dt. In this case, the key difference is that the velocity and position limits computed in Algorithm 5 in Sect. 2.3 are no longer sufficient to ensure compliance with the joint limits. To highlight this observation, we will derive the acceleration constraints from the position limits stated in (20). It can be demonstrated that for large sampling times dt, these constraints are more stringent compared to the ones obtained using the viability/invariance approach presented in Algorithm 4. Recalling (2), when considering $t \in [0, dt]$, the joint position $q_i$ reaches its extreme, such as the maximum $q_{M,i}$, when its time derivative is zero. i.e., $\frac{\partial q_i}{\partial t} = \dot{q}_i + t\ddot{q}_i = 0$. The time instant $t_{\text {max}}$ at which the maximum is reached is given by

$$\begin{aligned} t_{max} = -\frac{\dot{q}_i}{\ddot{q}_i}, \quad t \in [0, dt]. \end{aligned}$$

(33)

If

$$\begin{aligned} \dot{q}_i > 0 \quad \wedge \quad \ddot{q}_i \le -\frac{\dot{q}_i}{\delta t}, \end{aligned}$$

(34)

then the maximum $q_M = q_i + t_{max}\dot{q}_i + \frac{1}{2}t_{max}^2\ddot{q}_i = q_i - \frac{\dot{q}_i^2}{2\ddot{q}_i}$ is reached inside the time step [0, dt], leading to the constraint:

$$\begin{aligned} \ddot{q}_i \le -\frac{\dot{q}_i^2}{2(q_{M,i} - q_i)}. \end{aligned}$$

(35)

Instead, if the conditions in (34) are not met, the maximum is reached at the boundaries of the time step, in particular, the constraint needs to be enforced in $t = dt$, therefore:

$$\begin{aligned} \ddot{q}_i \le \frac{2}{dt^2}(q_{M,i} -q_i - dt \dot{q}_i). \end{aligned}$$

(36)

A similar analysis can be performed for the minimum position. The blue regions in Fig. 4a represent the areas in the state space where the acceleration constraints take precedence over the ones derived from the viability/invariance approach. The dashed black line corresponds to the upper joint limit $q_{M,i} = 2.09$. The orange region beyond the viability limit is infeasible due to the viability/invariance constraints. The expansion of the blue region within the feasible area indicates that the position limits defined in (20) cannot be satisfied without imposing the conditions described by (35) and (36). Additionally, the acceleration constraints obtained from the alternative necessary condition (26) cannot be disregarded anymore, and it is necessary to incorporate them into the implementation. As the sampling time dt decreases, this region becomes progressively smaller, rendering the measures employed in [4] less significant. Figure 4b provides an illustration of the number of states sampled from the feasible region where the conditions (35) and (36) impose stricter constraints compared to those computed using Algorithm 4.

Fig. 4.

Study on Viability and position constraints as proposed in [4]