Estimation of model parameters in a dynamic system can be significantly improved with the choice of experimental trajectory. For general nonlinear dynamic systems, finding globally “best” trajectories is typically not feasible; however, given an initial estimate of the model parameters and an initial trajectory, we present a continuous-time optimization method that produces a locally optimal trajectory for parameter estimation in the presence of measurement noise. The optimization algorithm is formulated to find system trajectories that improve a norm on the Fisher information matrix (FIM). A double-pendulum cart apparatus is used to numerically and experimentally validate this technique. In simulation, the optimized trajectory increases the minimum eigenvalue of the FIM by three orders of magnitude, compared with the initial trajectory. Experimental results show that this optimized trajectory translates to an order-of-magnitude improvement in the parameter estimate error in practice.
In the robotics field, trajectory optimization and parameter identification algorithms have been developed for special classes of robotic systems. For serial robot arms and similarly connected systems, chain-based techniques and linear separation of parameters can be used @cite_12 @cite_36 . Techniques have also been adapted for parallel robots and manipulators @cite_5 @cite_3 @cite_27 . While these techniques perform well for the intended class of robots, we seek an algorithm that has the ability to work on general nonlinear systems, only requiring differentiability of the dynamics and some form of control authority.
@cite_36 @cite_3 @cite_27 @cite_5 @cite_12
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Direct methods for trajectory optimization are widely used for planning locally optimal trajectories of robotic systems. Many critical tasks, such as locomotion and manipulation, often involve impacting the ground or objects in the environment. Most state-of-the-art techniques treat the discontinuous dynamics that result from impacts as discrete modes and restrict the search for a complete path to a specified sequence through these modes. Here we present a novel method for trajectory planning of rigid-body systems that contact their environment through inelastic impacts and Coulomb friction. This method eliminates the requirement for a priori mode ordering. Motivated by the formulation of multi-contact dynamics as a Linear Complementarity Problem for forward simulation, the proposed algorithm poses the optimization problem as a Mathematical Program with Complementarity Constraints. We leverage Sequential Quadratic Programming to naturally resolve contact constraint forces while simultaneously optimizing a trajectory that satisfies the complementarity constraints. The method scales well to high-dimensional systems with large numbers of possible modes. We demonstrate the approach on four increasingly complex systems: rotating a pinned object with a finger, simple grasping and manipulation, planar walking with the Spring Flamingo robot, and high-speed bipedal running on the FastRunner platform. This paper describes a new approach to the parameterization of robot excitation trajectories for optimal robot identification. The trajectory parameterization is based on a combined Fourier series and polynomial functions. The coefficients of the Fourier series are optimized for minimal sensitivity of the identification to measurement disturbances, which is measured as the d-optimality criterion, taking into account motion constraints in joint and Cartesian space. This parameterization satisfies both the guarantees of convergence by adding terms and the matching of the boundary conditions. Application of the method for the identification of the CRS A465 industrial robot proves the validity of the proposed approach. Although the dynamic equations of motion of open-chained robot systems are well known, they are seldom taken into account during the planning of motions. In this work, we show that the dynamics of a robot can be used to produce motions that extend the payload capability beyond the limit set by traditional methods. In particular, we develop a point-to-point weightlifting motion planner for open-chained robots. The governing optimal control problem is converted into a direct, SQP parameter optimization in which the gradient is determined analytically. The joint trajectories are defined by B-spline polynomials along with a time-scale factor. The algorithm is applied to a Puma 762 robot, with its physical limitations incorporated into the formulation. The torque limits are formulated as soft constraints added into the objective function while the position and velocity limits are formulated as hard, linear inequality constraints, on the parameters. The solutions obtained with our algorithm extend the robot's payload capability while reducing the joint torques. Interestingly, nearly all the trajectories found pass through singular configurations, where large internal forces from the robot are applied to the payload and little torque is needed from the motors. Transferring solutions found by trajectory optimization to robotic hardware remains a challenging task. When the optimization fully exploits the provided model to perform dynamic tasks, the presence of unmodeled dynamics renders the motion infeasible on the real system. Model errors cannot be only a result of model simplifications, but also naturally arise when deploying the robot in unstructured and nondeterministic environments. Predominantly, compliant contacts and actuator dynamics lead to bandwidth limitations. While classical control methods provide tools to synthesize controllers that are robust to a class of model errors, such a notion is missing in modern trajectory optimization, which is solved in the time domain. We propose frequency-shaped cost functions to achieve robust solutions in the context of optimal control for legged robots. Through simulation and hardware experiments we show that motion plans can be made compatible with bandwidth limits set by actuators and contact dynamics. The smoothness of the model predictive solutions can be continuously tuned without compromising the feasibility of the problem. Experiments with the quadrupedal robot ANYmal, which is driven by highly compliant series elastic actuators, showed significantly improved tracking performance of the planned motion, torque, and force trajectories and enabled the machine to walk robustly on terrain with unmodeled compliance.