Frederik Debrouwere

Time-Optimal Path Following for Robots With Convex–Concave Constraints Using Sequential Convex Programming

Time-optimal path following considers the problem of moving along a predetermined geometric path in minimum time. In the case of a robotic manipulator with simplified constraints, a convex reformulation of this optimal control problem has been derived previously. However, many applications in robotics feature constraints such as velocity-dependent torque constraints or torque rate constraints that destroy the convexity. The present paper proposes an efficient sequential convex programming (SCP) approach to solve the corresponding nonconvex optimal control problems by writing the nonconvex constraints as a difference of convex (DC) functions, resulting in convex-concave constraints. We consider seven practical applications that fit into the proposed framework even when mutually combined, illustrating the flexibility and practicality of the proposed framework. Furthermore, numerical simulations for some typical applications illustrate the fast convergence of the proposed method in only a few SCP iterations, confirming the efficiency of the proposed framework.

Experimental Validation of Combined Nonlinear Optimal Control and Estimation of an Overhead Crane

Abstract This paper validates the combination of nonlinear model predictive control and moving horizon estimation to optimally control an overhead crane. Real-time implementation of this combined optimal control and estimation approach with execution times far below the sampling time was realized through the use of automatic code generation. Besides experiments that reflect good point-to-point performance, the approach showed to be good in disturbance rejection as well as in servo-tracking.

Time-optimal path following for robots with trajectory jerk constraints using sequential convex programming

Time-optimal path following considers the problem of moving along a predetermined geometric path in minimum time. In the case of a robotic manipulator a convex reformulation of this optimal control problem has been derived previously [1]. However, the bang-bang nature of the time-optimal trajectories results in near-infinite jerks in joint space and operational (Cartesian) space. For systems with un-modeled flexibilities, this usually results in excitation of the resonant frequencies, hence in unwanted vibrations and acceleration peaks, contributing to a tracking error. These vibrations can be reduced by imposing jerk constraints on the trajectory [2]. However, these jerk constraints destroy the convexity of the time-optimal control problem. The present paper proposes an efficient sequential convex programming (SCP) approach to solve the corresponding non-convex optimal control problem by writing the non-convex jerk constraints as a difference of convex (DC) functions. We illustrate the developed approach by means of experiments with a seven d.o.f. robot. Furthermore, numerical simulations illustrate the fast convergence of the proposed method in only a few SCP iterations, confirming the efficiency and practicality of the proposed framework.

Iterative learning control for optimal path following problems

In optimal path following problems the motion along a given geometric path is optimized according to a desired objective while accounting for the system dynamics and system constraints. In the case of time-optimal path following, for example, the system input to move along the geometric path in minimal time is computed. In practice however, due to model-plant mismatch, (i) the geometric path is not followed exactly, and (ii) the optimized trajectory might be suboptimal, or even infeasible for the true plant. Assuming that the system performs the task repeatedly, this paper proposes an iterative learning control approach to improve the path following performance. The proposed learning algorithm is experimentally validated for a time-optimal path following problem on an XY-table. The results show that the developed ILC approach improves both the execution time and the accuracy significantly.

Convex time-optimal robot path following with Cartesian acceleration and inertial force and torque constraints:

In time-optimal robot path following, a predetermined geometric trajectory is followed exactly in a time-optimal way considering system constraints, for example, actuator constraints. For a simplified robotic manipulator, this optimization problem can be reformulated into a convex optimization problem when only considering some system constraints. In this article, the convex approach is extended to account for Cartesian acceleration constraints and in turn account for inertial forces and torques acting on a load held by the robot. The focus of this article is on the reformulation of these Cartesian acceleration and inertial forces and torques to preserve the convexity of the optimization problem. We present a series of applications that result in solving a convex optimization problem, illustrating the practicality of the proposed reformulations.

Time-optimal path following for robots with object collision avoidance using lagrangian duality

Time-optimal path following considers the problem of moving along a predetermined geometric path in minimum time while respecting system constraints. This paper focusses on time-optimal path following problems in robotics where collision must be avoided with other robots or moving obstacles. The developed method is based on the convex reformulation of the time-optimal path following problem with simplified dynamics presented in [1]. The robot and the obstacles are modelled as unions of convex polyhedra and the collision avoidance constraints are derived using Lagrangian duality. These constraints render the optimization problem non-convex. However, numerical simulations show that the resulting non-convex optimization problem can still be solved efficiently using a non-linear solver, due to the time-optimal path following formulation [1] and the proposed formulation of the collision avoidance constraints.

Time-optimal tube following for robotic manipulators

Time-optimal path following for robots considers the problem of moving along a predetermined Cartesian geometric path in minimum time. In practice this path need not be followed exactly, but within a certain tolerance; so that the motion may be executed faster. In this paper, we define this deviation as a tube around the given geometric path. This transforms the path following problem into a tube following problem. However, unlike the former, the latter is not convex.We propose a problem formulation that can still be solved efficiently, as illustrated by some numerical examples.

A sequential log barrier method for solving convex-concave problems with applications in robotics

In this paper we present a novel method to solve a convex-concave optimization problem. For this class of problems, several methods have already been developed. Sequential convex programming (SCP) is one of the state-of-the-art methods and involves solving a sequence of convex subproblems by linearizing the concave parts. These convex problems are e.g. solved by a log barrier method which solves a sequence of log barrier problems. To reduce the computational load we propose a sequential convex log barrier (SCLB) method where the main difference with SCP is that we search for an approximated solution of the convex subproblems by only solving one log barrier problem. We prove convergence of the proposed algorithm and we give some numerical examples that illustrate the decrease in computational load and similar convergence behaviour for a practical robotics application.

Optimal Tube Following for Robotic Manipulators

Abstract Optimal path following for robots considers the problem of moving along a predetermined Cartesian geometric end effector path (which is transformed into a predetermined geometric joint path), while some objective is minimized: e.g. motion time or energy loss. In practice it is often not required to follow a path exactly but only within a certain tolerance. By deviating from the path, within the allowable tolerance, one could gain in optimality. In this paper, we define the allowable deviation from the path as a tube around the given geometric path. We then search for the optimal motion inside the tube. This transforms the path following problem to a tube following problem. In contrast to the (time or energy) optimal path following problem, the tube following problem is not convex. However, we propose a problem formulation that can still be solved efficiently, as will be illustrated by some numerical examples.

Conterweight synthesis for time-optimal robotic path following

The goal of this research is to explore the potential of adding counterweights to the robot structure in order to decrease the optimal motion time for path following problems. Through the addition of counterweights to the robot links, the torques required to move the robot links could be compensated, and hence the actuator can use the excess of torque to move the link faster. The aim is hence to find the motion timing along a predetermined geometric path and simultaneously synthesise counterweights for the robot links. In general this is a strongly non-convex problem. However, by projecting the motion along the predetermined geometric path and transforming the dynamic parameters, a more efficient bi-linear optimization problem is obtained. This is illustrated with two numerical examples for basic robot systems, illustrating the proposed approach and relevance towards practical applications.