Yuval Tassa

MuJoCo: A physics engine for model-based control

We describe a new physics engine tailored to model-based control. Multi-joint dynamics are represented in generalized coordinates and computed via recursive algorithms. Contact responses are computed via efficient new algorithms we have developed, based on the modern velocity-stepping approach which avoids the difficulties with spring-dampers. Models are specified using either a high-level C++ API or an intuitive XML file format. A built-in compiler transforms the user model into an optimized data structure used for runtime computation. The engine can compute both forward and inverse dynamics. The latter are well-defined even in the presence of contacts and equality constraints. The model can include tendon wrapping as well as actuator activation states (e.g. pneumatic cylinders or muscles). To facilitate optimal control applications and in particular sampling and finite differencing, the dynamics can be evaluated for different states and controls in parallel. Around 400,000 dynamics evaluations per second are possible on a 12-core machine, for a 3D homanoid with 18 dofs and 6 active contacts. We have already used the engine in a number of control applications. It will soon be made publicly available.

Synthesis and stabilization of complex behaviors through online trajectory optimization

We present an online trajectory optimization method and software platform applicable to complex humanoid robots performing challenging tasks such as getting up from an arbitrary pose on the ground and recovering from large disturbances using dexterous acrobatic maneuvers. The resulting behaviors, illustrated in the attached video, are computed only 7 × slower than real time, on a standard PC. The video also shows results on the acrobot problem, planar swimming and one-legged hopping. These simpler problems can already be solved in real time, without pre-computing anything.

Control-limited differential dynamic programming

Trajectory optimizers are a powerful class of methods for generating goal-directed robot motion. Differential Dynamic Programming (DDP) is an indirect method which optimizes only over the unconstrained control-space and is therefore fast enough to allow real-time control of a full humanoid robot on modern computers. Although indirect methods automatically take into account state constraints, control limits pose a difficulty. This is particularly problematic when an expensive robot is strong enough to break itself. In this paper, we demonstrate that simple heuristics used to enforce limits (clamping and penalizing) are not efficient in general. We then propose a generalization of DDP which accommodates box inequality constraints on the controls, without significantly sacrificing convergence quality or computational effort. We apply our algorithm to three simulated problems, including the 36-DoF HRP-2 robot. A movie of our results can be found here goo.gl/eeiMnn.

An integrated system for real-time model predictive control of humanoid robots

Generating diverse behaviors with a humanoid robot requires a mix of human supervision and automatic control. Ideally, the users input is restricted to high-level instruction and guidance, and the controller is intelligent enough to accomplish the tasks autonomously. Here we describe an integrated system that achieves this goal. The automatic controller is based on real-time model-predictive control (MPC) applied to the full dynamics of the robot. This is possible due to the speed of our new physics engine (MuJoCo), the efficiency of our trajectory optimization algorithm, and the contact smoothing methods we have developed for the purpose of control optimization. In our system, the operator specifies subtasks by selecting from a menu of predefined cost functions, and optionally adjusting the mixing weights of the different cost terms in runtime. The resulting composite cost is sent to the MPC machinery which constructs a new locally-optimal time-varying linear feedback control law once every 30 msec, while planning 500 msec into the future. This control law is evaluated at 1 kHz to generate control signals for the robot, until the next control law becomes available. Performance is illustrated on a subset of the tasks from the DARPA Virtual Robotics Challenge.

Stochastic Differential Dynamic Programming

Although there has been a significant amount of work in the area of stochastic optimal control theory towards the development of new algorithms, the problem of how to control a stochastic nonlinear system remains an open research topic. Recent iterative linear quadratic optimal control methods iLQG handle control and state multiplicative noise while they are derived based on first order approximation of dynamics. On the other hand, methods such as Differential Dynamic Programming expand the dynamics up to the second order but so far they can handle nonlinear systems with additive noise. In this work we present a generalization of the classic Differential Dynamic Programming algorithm. We assume the existence of state and control multiplicative process noise, and proceed to derive the second-order expansion of the cost-to-go. We find the correction terms that arise from the stochastic assumption. Despite having quartic and cubic terms in the initial expression, we show that these vanish, leaving us with the same quadratic structure as standard DDP.

Stochastic Complementarity for Local Control of Discontinuous Dynamics

We present a method for smoothing discontinuous dynamics involving contact and friction, thereby facilitating the use of local optimization techniques for control. The method replaces the standard Linear Complementarity Problem with a Stochastic Linear Complementarity Problem. The resulting dynamics are continuously differentiable, and the resulting controllers are robust to disturbances. We demonstrate our method on a simulated 6-dimensional manipulation task, which involves a finger learning to spin an anchored object by repeated flicking.

Simulation tools for model-based robotics: Comparison of Bullet, Havok, MuJoCo, ODE and PhysX

There is growing need for software tools that can accurately simulate the complex dynamics of modern robots. While a number of candidates exist, the field is fragmented. It is difficult to select the best tool for a given project, or to predict how much effort will be needed and what the ultimate simulation performance will be. Here we introduce new quantitative measures of simulation performance, focusing on the numerical challenges that are typical for robotics as opposed to multi-body dynamics and gaming. We then present extensive simulation results, obtained within a new software framework for instantiating the same model in multiple engines and running side-by-side comparisons. Overall we find that each engine performs best on the type of system it was designed and optimized for: MuJoCo wins the robotics-related tests, while the gaming engines win the gaming-related tests without a clear leader among them. The simulations are illustrated in the accompanying movie.

Infinite-Horizon Model Predictive Control for Periodic Tasks with Contacts.

We present a method that combines offline trajectory optimization and online Model Predictive Control (MPC), generating robust controllers for complex periodic behavior in domains with unilateral constraints (e.g., contact with the environment). MPC offers robust and adaptive control even in high-dimensional domains; however, the online optimization gets stuck in local minima when the domains has discontinuous dynamics. Some methods of trajectory optimization that are immune to such problems, but these are often too slow to be applied online. In this paper, we use offline optimization to find the limit-cycle solution of an infinite-horizon average-cost optimal-control task. We then compute a local quadratic approximation of the Value function around this limit cycle. Finally, we use this quadratic approximation as the terminal cost of an online MPC. This combination of an offline solution of the infinite-horizon problem with an online MPC controller is known as Infinite Horizon Model Predictive Control (IHMPC), and has previously been applied only to simple stabilization objectives. Here we extend IHMPC to tackle periodic tasks, and demonstrate the power of our approach by synthesizing hopping behavior in a simulated robot. IHMPC involves a limited computational load, and can be executed online on a standard laptop computer. The resulting behavior is extremely robust, allowing the hopper to recover from virtually any perturbation. In real robotic domains, modeling errors are inevitable. We show how IHMPC is robust to modeling errors by altering the morphology of the robot; the same controller remains effective, even when the underlying infinite-horizon solution is no longer accurate.

Least Squares Solutions of the HJB Equation With Neural Network Value-Function Approximators

In this paper, we present an empirical study of iterative least squares minimization of the Hamilton-Jacobi-Bellman (HJB) residual with a neural network (NN) approximation of the value function. Although the nonlinearities in the optimal control problem and NN approximator preclude theoretical guarantees and raise concerns of numerical instabilities, we present two simple methods for promoting convergence, the effectiveness of which is presented in a series of experiments. The first method involves the gradual increase of the horizon time scale, with a corresponding gradual increase in value function complexity. The second method involves the assumption of stochastic dynamics which introduces a regularizing second derivative term to the HJB equation. A gradual reduction of this term provides further stabilization of the convergence. We demonstrate the solution of several problems, including the 4D inverted-pendulum system with bounded control. Our approach requires no initial stabilizing policy or any restrictive assumptions on the plant or cost function, only knowledge of the plant dynamics. In the appendix, we provide the equations for first- and second-order differential backpropagation.

Iterative local dynamic programming

We develop an iterative local dynamic programming method (iLDP) applicable to stochastic optimal control problems in continuous high-dimensional state and action spaces. Such problems are common in the control of biological movement, but cannot be handled by existing methods. iLDP can be considered a generalization of Differential Dynamic Programming, inasmuch as: (a) we use general basis functions rather than quadratics to approximate the optimal value function; (b) we introduce a collocation method that dispenses with explicit differentiation of the cost and dynamics and ties iLDP to the Unscented Kalman filter; (c) we adapt the local function approximator to the propagated state covariance, thus increasing accuracy at more likely states. Convergence is similar to quasi-Netwon methods. We illustrate iLDP on several problems including the “swimmer” dynamical system which has 14 state and 4 control variables.