Optimal Control for Clutched-Elastic Robots: A Contact-Implicit Approach

TL;DR

This work tackles energy-timing control in clutched-elastic robots by introducing a contact-implicit optimal control framework that jointly optimizes both the control input and the clutch sequence. By treating clutch torque as a freely adjustable decision variable and adding a penalty to discourage unnecessary switching, the method automatically discovers effective mode sequences without predefinition. Validation on a 2-DOF double pendulum with two Bi-Stiffness Actuators demonstrates up to a 30% improvement in end-effector speed over a guessed sequence, and real-world tracking with an LQR-based hybrid controller confirms robustness to model mismatch. The approach offers a principled, scalable path to harness elastic energy storage/release in CERs for dynamic, energy-efficient motions in multi-DoF platforms.

Abstract

Intrinsically elastic robots surpass their rigid counterparts in a range of different characteristics. By temporarily storing potential energy and subsequently converting it to kinetic energy, elastic robots are capable of highly dynamic motions even with limited motor power. However, the time-dependency of this energy storage and release mechanism remains one of the major challenges in controlling elastic robots. A possible remedy is the introduction of locking elements (i.e. clutches and brakes) in the drive train. This gives rise to a new class of robots, so-called clutched-elastic robots (CER), with which it is possible to precisely control the energy-transfer timing. A prevalent challenge in the realm of CERs is the automatic discovery of clutch sequences. Due to complexity, many methods still rely on pre-defined modes. In this paper, we introduce a novel contact-implicit scheme designed to optimize both control input and clutch sequence simultaneously. A penalty in the objective function ensures the prevention of unnecessary clutch transitions. We empirically demonstrate the effectiveness of our proposed method on a double pendulum equipped with two of our newly proposed clutch-based Bi-Stiffness Actuators (BSA).

Figures (5)

• Figure 1: Hardware setup (a): Experimental 2 DoF elastic pendulum actuated by two Bi-Stiffness Actuators. Internal structure (b): For each joint $j$, the spring inertia can be locked in place by brake $b_j$ (violet) and/or coupled directly to the link by clutch $c_j$ (cyan).
• Figure 2: Optimization results - Speed maximization at fixed final time. The upper and lower rows show the “guessed” and optimized sequence, respectively. The background colour indicates the operating mode (yellow for STG and turquoise for SEA)
• Figure 3: Experimental results - Speed maximization at fixed final time -- Guessed Sequence. The upper and lower rows show the results with and without LQR controller. The background colour indicates the operating mode (yellow for STG and turquoise for SEA). In the left and middle plots, you can observe the changes in motor angle $\dot{\theta}_{1,2}$ (red) and link angle $\dot{q}_{1,2}$ (blue) as raw and filtered data and the reference $\dot{q}_{1,2}^*$. The data smoothing procedure involved the application of robust quadratic regression, performed utilizing Matlab's smoothdata function. The plots on the right show the magnitude of the end effectors velocity $\vert v_{EE} \vert$ (blue) and acceleration $\vert a_{EE} \vert$ (red).
• Figure 4: Experimental results - Speed maximization at fixed final time -- Optimized Sequence. The plots are arranged as in Fig. \ref{['fig:exp_guess']}.
• Figure 5: Clutch torques generated by contact-implicit optimal control. The torques correspond to the STG and SEA modes, indicated by the background colour.