Towards Bipedal Behavior on a Quadrupedal Platform Using Optimal Control

This paper explores the applicability of a Linear Quadratic Regulator (LQR) controller design to the problem of bipedal stance on the Minitaur [1] quadrupedal robot. Balancing the body on only the rear legs affords the possibility of using the front legs for other tasks such as manipulation or bracing. Restricted to the sagittal plane, this behavior exposes a 3DOF (degree of freedom) double inverted pendulum with extensible length actuated at the prismatic (“shank”) and second revolute (“knee/hip”) joints per Figure 1 and section 2.1. Locking the prismatic (shank) DOF reduces the Lagrangian model of the pinned toe mechanism to that of the familiar singularly-actuated 2 DOF (revolute-revolute) acrobot per section 2.2 and lemma 2.1 . Since the linearized 3DOF dynamics at any vertically erect stance exposes a decoupled linearized acrobot, we focus on stabilizing this subsystem in isolation per section 2.3. Previous work has documented the empirical stabilization of a physical acrobot using a local LQR controller[2,3]. However, MATLAB simulations reveal that an LQR design very similar to those discussed in the past literature cannot achieve an empirically viable controller for our physical plant. First, the design is not robust, failing to stabilize the Minitaur model in numerical simulations run with a variety of small inaccuracies in presumed kinematic and dynamic parameters (none greater than than 5% of the putative value), even when the system starts at rest nearly exactly in the desired erect vertical equilibrium state per section 3.1.1 and tables 1 and 2. The fragility of the LQR design is similarly manifest when run on the Minitaur platform model whose initial prismatic shank extension length differs by an amount less 0.4% of its full range, and with initial body and leg angle conditions whose deviation relative to the desired erect vertical equilibrium goal state also lies beneath the sensor resolution threshold, per 3.1.2, fig. 2. Even worse, the acrobot physically instantiated by locking the actual Minitaur’s leg extension has an (unactuated) stance toe angle lacking a joint sensor so that its (intertial frame) body orientation must actually be inferred from IMU-driven estimators whose resolution (in both space and time) is considerably worse than that of its available (actuated) joint sensors (whereas the joint encoders offer 9 bits of useful resolution, the IMU position estimates have a noise floor of ±3◦-just under 1% of their 360◦ range) per 3.1.3, and figs 4, 3. Experimental studies using derivative measurments from the filtered IMU output show that the commensurately lagging joint velocity filter time constants cannot support the derivative gains required by the LQR design (per footnote †). Suitably lowered derivative gain magnitudes found by ad hoc tuning ∗ resulted in successful stabilization of the numerical acrobot model with physically realizable sensor estimation and actuator gains per section 3.1.4 and fig. 5. Unfortunately, further numerical study of the nonlinear Acrobot model revealed that the basin of attraction around the vertically erect equilibrium afforded by these physically realizable gains lay nearly below the physical (IMU-driven) sensor noise floor. This paper reports on experiments with two different approximate variants of the physical Minitaur platform that corroborate and underscore the implications of the simulation study. On a physical instance of the literal Acrobot scaled to minitaur proportions, equipped with high resolution (9-bit) joint sensors at both revolute (toe and hip/knee) joints, we successfully implemented both the local LQR design as well as its ad hoc relaxation Further author information: (Send correspondence to T.T. Topping.) T.T.Topping: E-mail: [email protected] V. Vasilopoulos.: E-mail: [email protected] ∗ Lowered-gains magnitudes can of course be achieved by ad hoc adjustment of the LQR cost structure, but our repeated efforts along these lines led to controllers whose sensitivity to parametric error was an order of magnitude worse than reported above. (with diminished velocity gains) albeit the latter only from initial conditions extraordinarily carefully placed at the targeted physically erect equilibrium state 3.2.1, fig. 9. We then attempted to run the same experiments on a second variant: a literal Minitaur platform that we modified by pinning its toes along a fixed axis, thereby forming a literal revolute first joint. † As predicted by our numerical study, neither the original LQR design, nor its velocity-gain relaxation succeeded in stabilizing the toe-pinned but IMU-driven Minitaur’s vertically erect equilibrium state per section 3.2.2 and figure 10. Finally, for completeness, an unmodified Minitaur model was tested to confirm the failure of the LQR based controller for bipedal stance 3.2.3, fig 11. We conclude that local LQR-based linearized controller designs are too fragile to stabilize the physical Minitaur platform around its vertically erect equilibrium and end with a brief assessment of a variety of more sophisticated nonlinear control approaches whose pursuit represents work presently in progress.