Abstract. The time optimal path tracking for industrial robots regards the problem of generating trajectories that follow predefined end-effector (EE) paths in shortest time possible taking into account kinematic and dynamic constraints. The complicated tasks used in industrial applications lead to very long EE paths. At the same time smooth trajectories are mandatory in order to increase the service life.The consideration of jerk and torque rate restrictions, necessary to achieve smooth trajectories, causes enormous numerical effort, and increases computation times. This is in particular due to the high number of optimization variables required for long geometric paths. In this paper we propose an approach where the path is split into segments. For each individual segment a smooth time optimal trajectory is determined and represented by a spline. The overall trajectory is then found by assembling these splines to the solution for the whole path. Further we will show that by using splines, the jerks are automatically bounded so that the jerk constraints do not have to be imposed in the optimization, which reduces the computational complexity. We present experimental results for a six-axis industrial robot. The proposed approach provides smooth time optimal trajectories for arbitrary long geometric paths in an efficient way.

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This paper presents a dynamic programming approach for calculating time optimal trajectories for industrial robots, subject to various physical constraints. In addition to path velocity, motor torque, joint velocity and acceleration constraints, the present contribution also shows how to deal with torque derivative and joint jerk limitations. First a Cartesian path for the endeffector is defined by splines using Bernstein polynomials as basis functions and is parameterized via a scalar path parameter. In order to compute the belonging quantities in configuration space, inverse kinematics is solved numerically. Using this and in combination with the dynamical model, joint torques as well as their derivatives can be constrained. For that purpose the equations of motion are calculated with the help of the Projection Equation. As a consequence of the used optimization problem formulation, the dynamical model as well as the restrictions have to be transformed to path parameter space. Due to the additional consideration of jerk and torque derivative constraints, the phase plane is expanded to a phase space. The parameterized restrictions lead to feasible regions in this space, in which the optimal solution is sought. Result of the optimization is the time behavior of the path parameter and subsequently the feed forward torques for the optimal motion on the spatial path defined by previously mentioned splines. Simulation results as well as experimental results for a three axes industrial robot are presented.

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Time-optimal path following, i.e. moving a robot's end-effector optimally along a specified geometric path, is a very important and well discussed problem in robotics. Nevertheless, most of the existing approaches concerning this topic neglect the speed dependency of torque constraints. The present paper presents a method for taking such constraints into account within a dynamic programming approach. To this end, the problem is treated in parameter space. This allows for an optimal use of existing resources. Due to the demanding constraints, precise mathematical models of the robots are indispensable. A satisfying match between model and real system can usually be achieved by parameter identification. For this purpose, it is a common way to derive the equations of motion using nominal parameters (masses, position of center of gravity, inertia and friction parameters), rewrite the equations in terms of linearly independent base parameters, and determine them with the help of measurements. Nevertheless, a parametrization of the motor torques has to be introduced in order to be able to consider their constraints within the optimization. In contrast to this, we present a general toolchain, based on the Projection Equation that directly derives the base parameter representation and furthermore the coefficients of the parametrized equations of motion. A verification in terms of a numerical example for a six-axis industrial robot demonstrate why speed dependent torque constraints are preferable over constant torque constraints for time-optimal robot trajectories. With a subsequent QR-decomposition, linearly independent base parameters p B are determined, resulting in Q = Θ B p B . In doing so, Θ B = ΘF Θ contains the linearly independent columns of Θ that are selected with the binary selection matrix F Θ . Numerical values for p B are then identified using a least squares error minimization.Parametrization of the EoM: Since the path is defined as a function of s, the EoM have to be parametrized along this path. The goal is to derive coefficients a Q (s), b Q (s), c Q (s) and d Q (s) so that the generalized torques can be written asA common approach is the insertion of specific values for q,q,q and g into Q = Θ(q,q,q)p, see e.g. [3]. This procedure has some drawbacks especially for systems with many DoF, since the resulting terms are not always easy to simplify. Alternatively, the parametrization of the EoM can elegantly be performed analytically using the previous mentioned method. For this reasonq(s) andq(s) are substituted into Θ Rc , Θ Rv , Θ T M and Θ T , where a separation concerning z , z and √ z is done. For indicating corresponding values an additional index is introduced. Terms concerning z are indexed with a, terms concerning z with b, terms concerning √ z with d and residual terms with c. After selecting the linearly independent columns of the parametrized information matrix with F Θ ,the coefficients follow by separation toThe matrices that are necessary for their calculation are mainly known from dynamic m...