Cartesian Path Research Articles

Generation of Configuration Space Trajectories Over Semi-Constrained Cartesian Paths for Robotic Manipulators

Serial-link manipulators are required to execute trajectories that enable a robot end-effector or a tool to track a Cartesian path. Practical applications may not require constraining all six degrees of freedom (position and orientation) of the tool resulting in semi-constrained paths. Semi-constrained paths allow improved success rates and better quality trajectories as the robot has more freedom to meet the kinematic and dynamic constraints. Additionally, robotic applications will need to use multiple tool center points (TCPs) on the tool to generate feasible paths for the robot. We present an iterative graph construction method to find trajectories for semi-constrained Cartesian paths that also use multiple TCPs. Our graph-based method finds multiple inverse kinematic solutions for possible Cartesian poses that the robot can take and connects them to build a graph. The algorithm uses cues from the Cartesian space to prioritize poses that produce a better quality solution. A biasing scheme is also developed to selectively sample the starting Cartesian poses from available choices. Our method finds near-optimal solutions with significantly fewer nodes and edges in the graph. The algorithm’s performance results on complex industrial test cases are provided. Note to Practitioners—Industrial applications permit the relaxation of one or more degrees of freedom of the tool while following the Cartesian paths. The relaxation of constraints is introduced by defining tolerances between the tool and the workpiece. Multiple TCPs have to be employed for many tasks to use different surfaces of the tool. In this paper, we present a planning algorithm for semi-constrained Cartesian paths that can incorporate the use of multiple TCPs. The user can define discrete TCPs over the tool contact points or surfaces. The tolerances can be easily defined as angular limits on tool orientation along the Cartesian path. Our planning algorithm can work with others via point constraints in Cartesian space or joint space of the robot. Practitioners from the industry can use the method presented in this paper to develop automated robotic cells that do cutting, sanding, polishing, welding, painting, composite layup, additive manufacturing, and several other common applications.

Method for the Calculation of Local Bead Volume in Multi-Axis Additive Manufacturing

In recent history, Multi-Axis Additive Manufacturing enjoys increasing attention for both industrial and research application. In comparison to traditional processes, the additional rotation axes enable parts to be printed with greater design flexibility, while reducing the need for support structures. The multi-axis motion additionally allows for optimization of structural properties of the parts. However, curved-layer printing requires significantly higher effort for planning and computation, as the layers exhibit curvature in their shape and feature local variations of the layer height and bead width. Consideration of these properties is often neglected in the development of path planning methods, although they pose a significant influence to the optical and structural quality of the printed parts. Local over- or under-extrusion can result in material accumulation or voids within the part and need to be avoided. Additionally, some Multi-Axis Additive Manufacturing processes like Directed Energy Deposition are especially prone to errors in practice, as the melt-pool behavior is hard to model and predict accurately. This paper introduces a method for the precise calculation and adjustment of local bead volume, as a possible solution to these problems. The method processes existing Cartesian paths and uses a model of the bead geometry to simulate the extrusion by rasterization. The method considers printing head orientation, as well as neighborhood information of adjacent beads and builds a volumetric map of the paths. Local extrusion volume is then determined for each waypoint of the path from this volume map and provided to the printer along with the G-Code, enabling adjusted volume deposition both in advance and during the process by iterative path-adaptation. The provided implementation is able to achieve favorable results for both use-cases but proves especially beneficial for the latter.

Efficient Inverse Kinematics and Planning of a Hybrid Active and Passive Cable-Driven Segmented Manipulator

A cable-driven segmented manipulator (CDSM) has superior dexterity for operations in confined space due to its light-slender body and redundant degree of freedoms (DOFs). However, its inverse kinematics resolving and configuration planning are very challenging due to the complex structure and strict constraints. In this article, we propose a two-layer geometric iteration (TLGI) method for inverse kinematics resolving and configuration-constrained Cartesian path planning. The computation efficiency is largely improved and singularities are avoided. First, the end-effector attitude is decomposed into a direction vector and a rotation angle. The former and the end-effector position are combined into state variables of the inner layer, and the latter is treated separately as the state variable of the outer layer. Then, the TLGI method enables to rapidly reach the desired 6-DOF pose by two-layer iterations, i.e., the inner and outer loop iteration. Second, during the inner loop iteration, the CDSM is modeled as an equivalent articulated arm whose end-effector position and direction is the same as that of CDSM, but its links length and joint angles depend on the current configuration of CDSM. Then, the efficient forward and backward reaching inverse kinematics (FABRIKs) method is extended to apply on CDSM so that it can fast reach the inner state variables. During the outer loop iteration, three different rotation cases, i.e., the rotating around the end, root, and both end and root, are designed to switch automatically to reach the outer state variable iteratively. Moreover, by parameterizing geometric constraints of the environment, a TLGI-based configuration-pose simultaneous planning method is also put forward to efficiently achieve additional configuration constraints for operations of CDSM in confined space. Finally, the proposed method is verified by both the simulations and experiments.

A Deep Reinforcement-Learning Approach for Inverse Kinematics Solution of a High Degree of Freedom Robotic Manipulator

The foundation and emphasis of robotic manipulator control is Inverse Kinematics (IK). Due to the complexity of derivation, difficulty of computation, and redundancy, traditional IK solutions pose numerous challenges to the operation of a variety of robotic manipulators. This paper develops a Deep Reinforcement Learning (RL) approach for solving the IK problem of a 7-Degree of Freedom (DOF) robotic manipulator using Product of Exponentials (PoE) as a Forward Kinematics (FK) computation tool and the Deep Q-Network (DQN) as an IK solver. The selected approach is architecturally simpler, making it faster and easier to implement, as well as more stable, because it is less sensitive to hyperparameters than continuous action spaces algorithms. The algorithm is designed to produce joint-space trajectories from a given end-effector trajectory. Different network architectures were explored and the output of the DQN was implemented experimentally on a Sawyer robotic arm. The DQN was able to find different trajectories corresponding to a specified Cartesian path of the end-effector. The network agent was able to learn random Bézier and straight-line end-effector trajectories in a reasonable time frame with good accuracy, demonstrating that even though DQN is mainly used in discrete solution spaces, it could be applied to generate joint space trajectories.

An Interference Inspection Algorithm of Limb and Hexapod Frame in the Treatment of Lower Limb Deformity

Pre-operative correction trajectory planning is one of the important aspects of deformity correction. Avoidance of limb-frame interference is essential to verify the implementability of the preplanned correction trajectory, as well as to maintain the continuity and security of the correction strategy. In this study, a novel interference inspection algorithm is developed to investigate the interaction of the limb and hexapod frame in the treatment of lower limb deformities. The algorithm is built on a minimum distance model of the cone frustum busbar and cylindrical axis using vector analysis. A predefined trajectory is generated by Cartesian coordinate path control. Subsequently, an interference case is performed through numerical simulation and motion simulation. The results show that the conclusion of numerical simulation and motion simulation is consistent, which prove the feasibility of the algorithm. The results also show that it is possible to identify the riskiest struts, which are prone to interfere with the limb, and the riskiest positions. The proposed algorithm can support the clinician in selecting the suitable frame configuration to avoid interference. The algorithm solves the problem that the interference can only be judged by clinical observation in the clinic.

Robust design of independent joint control of industrial robots with secondary encoders

The independent joint control approach with motor-side velocity and position feedback still dominates most industrial robot applications. Although it is known, that the elasticity of the drivetrains has a significant influence on the achievable dynamic path accuracy. As a result, an increasing number of industrial robots are available that use secondary encoders to measure the joint position. This paper addresses the robust design of independent joint control of industrial robots with secondary encoders under explicit consideration of the drivetrain elasticity. For this purpose, the nonlinear robot dynamics are linearized and the configuration dependency of the mass matrix is treated as structured uncertainty. The trade-off between the velocity control bandwidth and the gain margin of the position control loop is solved by means of a nonsmooth optimization-based H∞ synthesis. The design method is validated for the three base joints of a KUKA KR210-2 industrial robot. Experimental results for linear Cartesian paths and a milling experiment are presented to confirm the approach.

Cartesian Constrained Stochastic Trajectory Optimization for Motion Planning

This paper presents novel extensions of the Stochastic Optimization Motion Planning (STOMP), which considers cartesian path constraints. It potentially has high usage in many autonomous applications with robotic arms, where preservation or minimization of tool-point rotation is required. The original STOMP algorithm is unable to use the cartesian path constraints in a trajectory generation because it works only in robot joint space. Therefore, the designed solution, described in this paper, extends the most important parts of the algorithm to take into account cartesian constraints. The new sampling noise generator generates trajectory samples in cartesian space, while the new cost function evaluates them and minimizes traversed distance and rotation change of the tool-point in the resulting trajectory. These improvements are verified with simple experiments and the solution is compared with the original STOMP. Results of the experiments show that the implementation satisfies the cartesian constraints requirements.

A novel 3D path following control framework for robots performing surface finishing tasks

We describe a novel 3D path following control framework for articulated robots in applications where constant speed travel along a path is desirable, such as robotic surface finishing tasks. Given the desired robot configuration sequence with a list of waypoints along a path, a trajectory optimization scheme based on direct collocation is proposed to determine the Cartesian path and the maximum constant translation speed that are dynamically feasible. Employing the Hermite–Simpson collocation method, a Cartesian path is developed that not only preserves the characteristics of the original motion sequence but also satisfies the physical requirements of the robot kinematics and dynamics. Since joint velocity control is quite common in many industrial robots, we consider a 3D kinematic control in the robot tool frame with control inputs as rate of change of orientation. The objective for the translation motion is to achieve constant speed along the path tangent direction, and that of the orientation control is to orient the robot properly based on the path provided by a converging path planner. We describe the optimization procedure employed with the direct collocation method to obtain the desired Cartesian path, an arc-length based re-parametrization of the desired path, and a path planner that provides a converging path to the desired path. To perform the surface finishing operation, we further present the joint space control law that is converted from the synthesis of the proposed path following and impedance force control in the tool frame. To verify and evaluate the performance of the proposed framework, we have conducted extensive experiments with a six degrees-of-freedom (DOF) industrial robot for several paths that can be employed for surface finishing of a variety of industrial parts.

A Real-Time-Capable Closed-Form Multi-Objective Redundancy Resolution Scheme for Seven-DoF Serial Manipulators

Using a closed-form inverse kinematics solution for motion planning has many advantages compared to traditional numerical approaches, most notably much faster computation times and better suitability for real-time applications. Steady progress has been made to develop an analytic inverse kinematics solution for seven degrees of freedom (DoF) manipulators without joint offsets. Redundancy resolutions have been proposed to push the individual joint angles away from their limits. However, no further objectives are considered in these closed-form solutions. We propose a multi-objective real-time optimization approach that is able to minimize joint velocities and accelerations, while still avoiding joint limits. A closed-form solution to the optimization is derived, which only requires a single intuitive tuning parameter. We provide a free and open-source software implementation of the full kinematics algorithm that covers the complete solution space. The null-space motion resulting from our method is evaluated, achieving faster motion times and smoother arm motions along Cartesian paths in comparison to the state-of-the-art approach. Finally, experimental results with a KUKA iiwa robot are reported.

An Experimentally Validated Technique for the Real-Time Management of Wrist Singularities in Nonredundant Anthropomorphic Manipulators

The automatic management of kinematic singularities that are typical for trajectories planned in the operational space is arousing a renewed interest among the scientific community because the most recent strategies make it possible their real-time management. The approach described in this paper allows executing trajectories in the operational space which pass through wrist singularities. It introduces several novelties with respect to known alternative strategies. First, it is conceived for trajectories which are planned on-the-fly. Second, singularities are avoided by changing slightly the tool frame orientation while strictly preserving both the assigned Cartesian path and time law. Finally, the approach is effective also for manipulators moving at standard operative speeds, and it explicitly handles given limits on joint velocities and accelerations. In this paper, an approach proposed in early works is revised in order to make it ready for an industrial implementation. In particular, a procedural method is proposed for the tuning of the algorithm so as to make it more deterministic and to increase the success rates. Furthermore, the singularity avoidance problem is theoretically analyzed in order to devise a necessary condition for the existence of a solution. Results are experimentally validated through an anthropomorphic industrial manipulator.

Minimum-time path planning for robot manipulators using path parameter optimization with external force and frictions

This paper presents a new minimum-time trajectory planning method which consists of a desired path in the Cartesian space to a manipulator under external forces subject to the input voltage of the actuators. Firstly, the path is parametrized with an unknown parameter called a path parameter. This parameter is considered a function of time and an unknown parameter vector for optimization. Secondly, the optimization problem is converted into a regular parameter optimization problem, subject to the equations of motion and limitations in angular velocity, angular acceleration, angular jerk, input torques of actuators’, input voltage and final time, respectively. In the presented algorithm, the final time of the task is divided into known partitions, and the final time is an additional unknown variable in the optimization problem. The algorithm attempts to minimize the final time by optimizing the path parameter, thus it is parametrized as a polynomial of time with some unknown parameters. The algorithm can have a smooth input voltage in an allowable range; then all motion parameters and the jerk will remain smooth. Finally, the simulation study shows that the presented approach is efficient in the trajectory planning for a manipulator that wants to follow a Cartesian path. In simulations, the constraints are respected, and all motion variables and path parameters remain smooth.

The Application of Noise-Tolerant ZD Design Formula to Robots’ Kinematic Control via Time-Varying Nonlinear Equations Solving

Recently, a new formula with noise-tolerant capability has been designed by Zhang et al. , and the resultant Zhang dynamics (ZD) models have been developed to solve different types of time-varying problems. Based on previous research, this paper presents and investigates the application of such a noise-tolerant ZD design formula to kinematic control of redundant robot manipulators via time-varying nonlinear equations solving. Specifically, by exploiting this ZD design formula to solve the system of time-varying nonlinear kinematic equations involved in robot control, the redundancy-resolution scheme is established. Such a scheme contains the proportional, integral, and derivative information of the desired Cartesian path of the robot end-effector and can thus be viewed as a nonlinear proportional-integral-derivative controller for redundant robot manipulators. Then, theoretical results are given to show that the redundancy-resolution scheme has the capability of noise suppressing. Simulation results based on a four-link planar robot manipulator and a PA10 robot manipulator with zero, constant, and bounded time-varying noises further substantiate the efficacy and superiority of the redundancy-resolution scheme, and show the application prospect of the noise-tolerant ZD design formula.

New Pseudoinverse-Based Path-Planning Scheme With PID Characteristic for Redundant Robot Manipulators in the Presence of Noise

In this paper, a new path-planning scheme based on pseudoinverse-type formulation for redundant robot manipulators with the existence of noise is proposed and investigated. Such a pseudoinverse-based path-planning (PPP) scheme contains the proportional, integral, and derivative information of the desired Cartesian path (of the end-effector), and can thus be viewed as a nonlinear proportional–integral–derivative (PID) controller for redundant robot manipulators. In other words, the proposed PPP scheme has the PID characteristic in terms of the desired Cartesian path. Theoretical results are given to show that the Cartesian error synthesized by the proposed PPP scheme has the property of global and exponential convergence (that is to say, the error trajectory is asymptotically stable) and to indicate that such a scheme has the capability to suppress noise. Simulation results based on a four-link planar robot manipulator with the existence of zero, constant, and bounded time-varying noises are presented to further substantiate the efficacy and superiority of the proposed PPP scheme for the path planning of redundant robot manipulators. The physical realizability of the proposed PPP scheme is also demonstrated by applying this scheme to a practical six-link planar robot manipulator.

Faster Motion on Cartesian Paths Exploiting Robot Redundancy at the Acceleration Level

The problem of minimizing the transfer time along a given Cartesian path for redundant robots can be approached in two steps, by separating the generation of a joint path associated to the Cartesian path from the exact minimization of motion time under kinematic/dynamic bounds along the obtained parameterized joint path. In this framework, multiple suboptimal solutions can be found, depending on how redundancy is locally resolved in the joint space within the first step. We propose a solution method that works at the acceleration level, by using weighted pseudoinversion, optimizing an inertia-related criterion, and including null-space damping. Several numerical results obtained on different robot systems demonstrate consistently good behaviors and definitely faster motion times in comparison with related methods proposed in the literature. The motion time obtained with our method is reasonably close to the global time-optimal solution along same Cartesian path. Experimental results on a KUKA LWR IV are also reported, showing the tracking control performance on the executed motions.

Cartesian Impedance Control for Physical Human–Robot Interaction Using Virtual Decomposition Control Approach

This paper presents and verifies a novel Cartesian impedance control of the flexible joint manipulator for physical human–robot interaction application based on virtual decomposition control approach. Firstly, the Cartesian impedance control based on virtual decomposition control (VDC) is presented, and the asymptotical stability of the controller is proven by Lyapunov stability theorem. Compared with traditional methods based on singular perturbation, this method can greatly reduce the computational loads and is more suitable for real-time application. Then, a Cartesian force-feedback path planning combined with Cartesian impedance control based on VDC was used to keep the real contact force within the desired value to protect the manipulator and objects as a force, position, velocity and acceleration sensor, and the sensor can be configured freely by regulating the stiffness, damping and inertia, so the manipulator can interact with human (or unknown environment) in a friendly manner. The experimental results illustrate the validity of the developed VDC-based impedance control approach.

GA Based Adaptive Singularity-Robust Path Planning of Space Robot for On-Orbit Detection

As a new on-orbit detection platform, the space robot could ensure stable and reliable operation of spacecraft in complex space environments. The tracking accuracy of the space manipulator end-effector is crucial to the detection precision. In this paper, the Cartesian path planning method of velocity level inverse kinematics based on generalized Jacobian matrix (GJM) is proposed. The GJM will come across singularity issue in path planning, which leads to the infinite or incalculable joint velocity. To solve this issue, firstly, the singular value decomposition (SVD) is used for exposition of the singularity avoidance principle of the damped least squares (DLS) method. After that, the DLS method is improved by introducing an adaptive damping factor which changes with the singularity. Finally, in order to improve the tracking accuracy of the singularity-robust algorithm, the objective function is established, and two adaptive parameters are optimized by genetic algorithm (GA). The simulation of a 6-DOF free-floating space robot is carried out, and the results show that, compared with DLS method, the proposed method could improve the tracking accuracy of space manipulator end-effector.

Joint-Space Kinematic Model for Gravity-Referenced Joint Angle Estimation of Heavy-Duty Manipulators

A gravity-referenced joint angle estimation, which acts in joint space, is proposed for multiple-degree-of-freedom hydraulic manipulators. As a novelty, the estimation pairs up inertial units across a section of an open kinematic chain. In this way, the three-axis linear accelerometers and the three-axis rate gyros provide a drift-free solution for observing the motion state of a rotary joint connecting two links without relying on the full forward kinematics. For a low-noise, low-delay estimate, the linear accelerations acting on the paired inertial units are modeled and robustly combined with the principles of complementary and Kalman filtering. In pick-and-place experiments with a serial-link manipulator on a multiton, off-road forestry vehicle, joint angle sensing error of less than ±1° was achieved in spite of the dynamic interaction between the vehicle base and the terrain. Furthermore, the kinematic modeling’s ability to compensate for the nonplanar, coupled 3-D linkage motion is studied for one- and two-axis rate measurements along with Cartesian path tracking. This gives new insights from the typical planar kinematic models and heavy-duty control viewpoints.

Analysis of reaction torque-based control of a redundant free-floating space robot

Owing to the dynamics coupling between a free-floating base and a manipulator, the non-stationary base of a space robot will face the issue of base disturbance due to a manipulator’s motion. The reaction torque acted on the satellite base’s centroid is an important index to measure the satellite base’s disturbance. In this paper, a comprehensive analysis of the reaction torque is made, and a novel way to derive the analytical form of the reaction torque is proposed. In addition, the reaction torque null-space is derived, in which the manipulator’s joint motion is dynamically decoupled from the motion of the satellite base, and its novel expression demonstrates the equivalence between the reaction torque null-space and the reaction null-space. Furthermore, the reaction torque acted as an optimization index can be utilized to achieve satellite base disturbance minimization in the generalized Jacobian-based end-effector Cartesian path tracking task. Besides, supposing that the redundant degrees of freedom are abundant to achieve reaction torque-based active control, the reaction torque can be used to realize satellite base attitude control, that is, base attitude adjustment or maintenance. Moreover, because reaction torque-based control is a second-order control scheme, joint torque minimization can be regarded as the optimization task in reaction torque-based active or in-active control. A real-time simulation system of a 7-DOF space robot under Linux/RTAI is developed to verify and test the feasibility and reliability of the proposed ideas. Our extensive empirical results demonstrate that the corresponding analysis about the reaction torque is correct and the proposed methods are feasible.

Time Optimal Path Planning for Industrial Robots: A Dynamic Programming Approach Considering Torque Derivative and Jerk Constraints

AbstractThis 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. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Continuous Genetic Algorithms for Collision-Free Cartesian Path Planning of Robot Manipulators

A novel continuous genetic algorithm (CGA) along with distance algorithm for solving collisions-free path planning problem for robot manipulators is presented in this paper. Given the desired Cartesian path to be followed by the manipulator, the robot configuration as described by the D-H parameters, and the available stationary obstacles in the workspace of the manipulator, the proposed approach will autonomously select a collision free path for the manipulator that minimizes the deviation between the generated and the desired Cartesian path, satisfy the joints limits of the manipulator, and maximize the minimum distance between the manipulator links and the obstacles. One of the main features of the algorithm is that it avoids the manipulator kinematic singularities due to the inclusion of forward kinematics model in the calculations instead of the inverse kinematics. The new robot path planning approach has been applied to two different robot configurations; 2R and PUMA 560, as non-redundant manipulators. Simulation results show that the proposed CGA will always select the safest path avoiding obstacles within the manipulator workspace regardless of whether there is a unique feasible solution, in terms of joint limits, or there are multiple feasible solutions. In addition to that, the generated path in Cartesian space will be of very minimal deviation from the desired one.