Among the many challenges of parallel kinematic manipulators, achieving high-speed and accurate control remains crucial. Estimating their dynamic properties is essential for designing precise and efficient control schemes. Conventional methods for dynamic model identification have been effective, though deep learning approaches have historically faced limitations due to data inefficiencies. However, recent advancements in physics-informed neural networks (PINNs) offer a way to improve both control and the extraction of interpretable physical properties from these robots. In this work, we propose and validate a PINN-based dynamic model for a Delta parallel robot, specifically the ABB IRB 360-6/1600. Our approach incorporates known physical properties, such as mass matrix sparsity, to improve accuracy and computational efficiency in dynamic model identification. To the best of our knowledge, this is the first study applying PINNs to model parallel robots. The method is validated experimentally, and its performance is compared to a validated identification technique for physically consistent identification, demonstrating the effectiveness of this approach for real-world applications in parallel robots.

The increased complexity of vehicle testing can be attributed to the rapid development of technical advancements within the automotive industry, thereby prolonging the time to market of a product. The process of allocating and coordinating vehicle tests at proving grounds (PGs) is a complex and time-consuming task. Currently, this process is still performed manually, which is inefficient. This study proposes a methodology for assigning scenarios to designated sites, taking into account travel aspects between locations and fulfilling participant requirements. The allocation procedure is formulated as an Open Job Shop Scheduling problem with temporal synchronisation and skill matching, and is solved by the Constraint Programming tool Google OR-Tools. The efficacy of the approach is demonstrated by its ability to generate a close-to-optimal schedule to fulfil customer requests. Case studies demonstrate that a combination of two distinct objectives are essential to meet the demands of compactness and time efficiency. The findings of this study provide a solid foundation for enhancing automation at a PG, thereby improving efficiency and optimising testing processes.

The determination of the dynamic properties of a robot is especially important for designing highly accurate and efficient control systems. Conventional methods for dynamic model identification have proven to be effective, where deep learning (DL) approaches have shown limits due to data inefficiencies. However, thanks to novel physics-informed DL architectures, such as Deep Lagrangian Networks (DeLaN) [1], it is possible to control and extract interpretable physical information of a robot. This paper introduces an augmented DeLaN architecture for linear viscous and non-symmetrical Coulomb friction identification, which also learns motor parameters such as rotor inertia. An approach is proposed for comparing this method with the conventional dynamic identification and previous DeLaN implementations. Moreover, our friction and rotor inertia identification is validated, and the performance of our model is analyzed with a real robot (UR5e).

Interpolation of rigid body motions, or a general frame motion in Euclidean space, is a recurring topic in robotics. It boils down to generating trajectories in a Lie group, either SE (3) or SO (3) × ℝ3, with given initial and/or terminal values. To this end, spline interpolation schemes were developed where canonical coordinates are represented by cubic splines. They allow for prescribing initial velocity and acceleration only. In many robotic applications, terminal conditions are prescribed, however. In this paper, a novel interpolation scheme is presented that admits prescribing the terminal pose, velocity and acceleration, or the initial condition. As example, the scheme is applied to a rendezvous task of a UAV and describing the deformation of a Cosserat beam as relevant for soft robotics. The presented interpolation scheme can be directly applied to the motion parameterization in terms of (dual) quaternions.

The kinematics of cuspidal 3R regional robots was studied extensively in the past. Moreover, certain industrial 6R robots were found to be cuspidal (e.g. Fanuc CRX series, Kinova GEN2), which makes cuspidal robots finally interesting for practical applications. This necessitates optimal trajectory planning, respecting the dynamics and technical limits of the particular robot. In this paper, a method for singularity-free time-optimal point-to-point trajectory (PtP) trajectory planning is proposed. As a special case, this method is applicable to time-optimal singularity-free assembly mode changing. Results are shown for 3R robots and a 6R Fanuc CRX10iA/L.

Parallel kinematics machines (PKM) operate with maximal acceleration being designed for highly dynamic manipulation tasks. This leads to extreme loads of the joints, which is usually not accounted for in the motion planning. In this paper an extended inverse dynamics method is introduced, which allows computing the joint reaction forces along with the actuation torques, and provides a basis for time optimal motion planning and control minimizing wear of the components. To this end, PKM are modeled using absolute coordinates. The joint constraints are complemented with servo constraints so that the motion can be described by the actuator motion or by the end-effector motion. The presented method is particularly advantageous when certain model parameters are unknown and allows for model simplification, which would not be possible for the relative coordinate formulation. The sparsity of the obtained velocity constraint Jacobian matrix, due to the use of absolute coordinates, can be efficiently exploited to minimize computation time. The method is demonstrated and numerical results are reported for a time-optimal pick and place movement of a 4-DOF Delta robot.

Modeling closed loop mechanisms is a necessity for the control and simulation of various systems and poses a great challenge to rigid body dynamics algorithms. Solving the forward and inverse dynamics for such systems require resolution of loop closure constraints which are often solved via numerical procedures. This brings an additional burden to these algorithms as they have to stabilize and control the loop closure errors. In order to avoid this issue, analytical solutions are preferred for commonly studied parallel mechanisms. This paper has two contributions. Firstly, it reports a case study on a modular and hybrid numerical-analytical approach to model and control series-parallel hybrid robots which are subjected to large number of holonomic constraints. The approach exploits the modularity in the robot design to combine analytical loop closure for the known submechanisms and numerical loop closure for submechanisms where analytical solutions are not available. This offers an edge over purely numerical approach in terms of computational efficiency. Secondly, an adaption of the constraint embedding approach in Articulated Body Algorithm (ABA) is presented which yields a recursive algorithm in minimal coordinates for computing the forward dynamics of series-parallel hybrid systems. The proposed modification exploits the Lie group formulations and allows easy implementation of recursive forward dynamics of constrained systems in state of the art multi-body solvers.

Derivatives of equations of motion describing the rigid body dynamics are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the equations of motion (EOM). This paper presents novel nthorder time derivatives of the EOM in both closed and recursive forms. While the former provides a direct insight into the structure of these derivatives, the latter leads to their highly efficient implementation for large degree of freedom robotic system.

In manufacturing industry, one of the main targets is to increase automation and ultimately to avoid failures under all circumstances. The plugging and locking of connectors is a class of tasks which is yet hard to be automatized with sufficiently high process stability. Due to the variation of plugging positions and external disturbances, e.g. occlusion due to cables, the quality assessment of plugging processes has emerged as a challenging task for image-based systems. For this reason, the proposed approach analyzes the inherent acoustic connector locking properties in combination with different neural network architectures in order to correctly identify connector locking signals and further to distinguish them from other machining events occurring in assembly plants. For this specific task, highly sensitive optical microphones have been applied for data acquisition. The proposed experiments are carried out under laboratory conditions as well as for the more complex situation in a real manufacturing environment. In this context, the usage of multimodal neural network architectures achieved highest levels in classification performance with accuracy levels close to 90%.

Quality assessment of products produced in small numbers requires discontinuous allocation of work forces. Automatizing these processes leads to a significant increase in cost efficiency. In this paper the testing of a mechatronic product is addressed, which requires in particular connecting it to a controller by means of an electric plug. The paper presents a robotic solution that allows to robustly accomplish the electric connection. The solution relies on a model-based hybrid force-position control of an industrial robot. The particular challenge is that the electric plug is multipolar, i.e. it must be inserted with a certain orientation. The latter necessitates a two stage approach for insertion, which distinguishes this problem from the classical peg-in-the-hole problem. Experimental results are presented for a prototype implementation on the real hardware. These results show a high success rate, and prove the practical applicability of the developed method.

It is becoming increasingly popular to use parallel mechanisms as modular subsystem units in the design of various robots for their superior stiffness, payload-to-weight ratio and dynamic properties. This leads to series-parallel hybrid robotic systems which pose several challenges in their modeling and control e.g. resolution of loop closure constraints, large size of their spanning tree etc. These robots are typically position-controlled and when equipped with real time dynamic control, often a simplified inverse dynamic model of these systems is utilized. However, the trade-offs of this model simplification has not been studied previously. This paper presents a representative study of the neglected dynamics by introducing some error metrics which are useful in highlighting the advantages and disadvantages of such model simplification. The study is guided with the help of a series-parallel humanoid leg which has been recently developed at DFKI-RIC.

This paper presents a new dynamic programming (DP) approach to the optimal path following problem. The method rests on an interpolation in the phase plane so that the resulting joint accelerations and joint torques are continuous. This allows for taking into account limits on the joint jerks and torque rates in addition to joint velocities, accelerations, torques, and the mechanical power. Most methods proposed in the literature yield values of optimal trajectories at discrete sampling times so that this must be interpolated subsequently to the trajectory optimization. This causes violations of the joint jerk and torque rate limits. The proposed method does not suffer from this problem, which is a main feature of this approach. Unlike most of the previously proposed methods, joint jerk and torque rate limits are addressed with a reasonable increase in the computation time of DP algorithms. Special attention is given to experimental validation of the optimization results. The presented experimental results confirm the importance of taking the motor torque characteristics as well as the Coulomb and viscous friction into account. Neglecting these effects (as most previous publications did) leads to trajectories that cannot be performed by real manipulators. Apart from DP, other approaches with smaller computation times exist. However, most of these methods are either limited to the time optimal case (which might not always be the desired criterion) and are not able to handle all earlier mentioned constraints or cannot take into account viscous friction effects. Apart from that a sequential convex programming (SCP) approach exists, which accounts for the same constraints as the presented approach. Therefore, the proposed DP approach is compared to this SCP method and an example is presented where the time optimal trajectory is performed by a real manipulator.

Motion planning for manipulators under task space constraints is difficult as it constrains the joint configurations to always lie on an implicitly defined manifold. It is possible to view task constrained motion planning as an optimization problem with non-linear equality constraints, which can be solved by general non-linear optimization techniques. In this paper, we present a novel custom optimizer which exploits the underlying structure present in many task constraints. At the core of our approach are some simple reformulations, which when coupled with the method of alternating projection, leads to an efficient convex optimization based routine for computing a feasible solution to the task constraints. We subsequently build on this result and use the concept of Augmented Lagrangian to guide the feasible solutions towards those that also minimize the user defined cost function. We show that the proposed optimizer is fully distributive and thus, can be easily parallelized. We validate our formulation on some common robotic benchmark problems. In particular, we show that the proposed optimizer achieves cyclic motion in the joint space corresponding to a similar nature trajectory in the task space. Furthermore, as a baseline, we compare the proposed optimizer with an off-the-shelf non-linear solver provide in open source package SciPy. We show that for similar task constraint residuals and smoothness cost, it can be upto more than three times faster than the SciPy alternative.

Time-optimal motion control will only find industrial applications if the optimal motions can actually be performed by standard industrial robots. This is not ensured by any optimal motion planning scheme proposed up to now. The limiting aspect rendering all these schemes impractical is the insufficient continuity of the motion trajectories. In this paper, a time-optimal path following along a predefined end-effector path is addressed for kinematically redundant robots, where nonredundant robots are included as special cases. As prerequisite explicit expressions for the higher order inverse kinematics are presented. Kinematic redundancy is resolved and exploited within the trajectory planning using the joint space decomposition and a novel pseudoinverse-based solution of the higher order inverse kinematics. The approaches are demonstrated for two examples of kinematically redundant manipulators performing time-optimal motions along prescribed end-effector paths in compliance with technological constraints. The optimization results are experimentally validated.

The classical optimal path following problem considers the problem of moving optimally along a predefined geometric path under technological restrictions. In contrast to optimal path following, optimal tube following allows deviations from the initial path within a predefined tube to reduce cost even more. The present paper proposes a modern approach that treats this non-convex problem in task space. This novel method also provides a simple way to derive optimal trajectories within a tube described in terms of polygonal lines. Numerical examples are presented that allow to compare the proposed method to existing joint space approaches.

Minimum-time trajectories for applications where a geometric path is followed by a kinematically redundant robot’s end-effector may yield economical improvements in many cases compared to conventional manipulators. While for non-redundant robots the problem of finding such trajectories has been solved, the redundant case has not been treated exhaustively. In this contribution, the problem is split into two interlaced parts: inverse kinematics and trajectory optimization. In a direct optimization approach, the inverse kinematics problem is solved numerically at each time point. Therein, the manupulator’s kinematic redundancy is exploited by introducing scaled nullspace basis vectors of the Jacobian of differential velocities. The scaling factors for each time point are decision variables, thus the inverse kinematics is solved optimally w.r.t. the trajectory optimization goal, i.e. minimizing end time. The effectiveness of the presented method is shown by means of the example of a planar 4R manipulator with two redundant degrees of freedom.

Vehicles with several centered orientable wheels have one of the highest maneuverability and are hence an excellent choice for transportation tasks in narrow environments. However, they are non-holonomic, in general redundantly actuated, and additionally suffer from configuration singularities, which makes their modeling and control challenging. Existing control approaches only consider the vehicle kinematics whereas the required torques are commonly controlled by classical PD motor controllers. However, this leads to considerable tracking errors and a violation of the constraints especially during acceleration phases. Moreover, actuator counteractions and an undefined torque distribution can be observed. This paper introduces a model-based control concept that overcomes these issues. It resolves counteractions and distributes torques according to physical limitations which significantly reduces slippage and the energy consumption and further reduces the tracking error. To this end, an inverse dynamics solution of a redundantly parametrized model is used. The method is robust to configuration singularities. This is confirmed by experimental results.

Nowadays robotic manipulators are used as multi-purpose tools and must be able to complete various tasks. Pure position control schemes are often not sufficient to fulfill the requirements of these tasks. Interaction with the environment requires an extension of the conventional position control in order to achieve a desired compliance, and thus to limit the impact in order to avoid damaging of involved objects. This paper presents an admittance control scheme applicable to kinematically redundant manipulators without using joint torque sensors or wrist-mounted force/torque sensors. By using motor current measurements an estimation of the external forces acting on the manipulator can be obtained and allows for a compliant behavior. Joint friction effects are overcome by superposing an additional movement in the null-space of the end-effector task. This control scheme is applied to an industrial manipulator, namely a Stäubli TX90L mounted on a linear axis (constituting a redundant 7-DOF manipulator) and experimental results are provided.

Minimum-time trajectories for applications where a geometric path is followed by a kinematically redundant robot's end-effector may yield economical improvements in many cases compared to conventional manipulators. While for non-redundant robots the problem of finding such trajectories has been solved, the redundant case has not been treated exhaustively. In this contribution, the problem is treated as two interdependent subproblems: inverse kinematics and trajectory optimization. Therein, a differential inverse kinematics resolution scheme is augmented by adding an optimal linear combination of nullspace basis vectors of the corresponding velocity Jacobian. Using the practical example of an industrial robot with 7 degrees of freedom performing a 5 degrees of freedom task, the effectiveness of the presented method is shown. Comparisons are made with a joint space decomposition inverse kinematics resolution approach.

Many robotics applications require smooth orientation planning, i.e. interpolation or approximation of a frame orientation through prescribed configurations such that the angular velocity and its time derivatives are smooth. This for instance ensures the continuity of the motor torques of robotic manipulator. Yet no satisfactory solution to this problem has been presented. This paper presents a solution using a B-spline parameterization of rotations. The method allows for a continuous interpolation and approximation through a given set of quaternions up to a specified order. The algorithm resembles the well-known B-spline interpolation in the sense that it boils down to solving a system of linear equations. The method is demonstrated for a polishing application (car fender) with an industrial robot.

The use of more than one steerable (standard) driving wheel allows a robot to perform omnidirectional motions. However, the modeling and control of such robots is challenging since the system is non-holonomic, nonlinear and typically over actuated. Moreover, such platforms exhibit kinematic singularities. A well known singular configuration is the configuration where two steerable driving wheels are coaxial aligned. This is highly problematic since this configuration corresponds to pure rotations, which is crucial for narrow space navigation. In this paper a control scheme with improved robustness w.r.t. these singularities is derived. It is based on the second order (accelerations) non-holonomic constraints. The remaining singularity is tackled by a regular parametrization of the robot's motion. Thereupon a novel control concept is presented which is based on an input-output linearization in terms of a path parameter. The choice of this parametrization provides an additional parameter in the controller design. The approach is demonstrated for a prototype implementation.

Decentralized individual control is still the state of the art in industrial applications. While redundantly actuated parallel kinematic machine (RA-PKM) possess desirable kinematic and dynamic properties, their control is impeded by the occurrence of antagonistic control forces. Such antagonistic forces are inherent to the decentralized control of RA-PKM that, together with calibration errors and finite encoder resolutions, causes antagonistic control forces and excited vibrations, and, hence, energy loss and instabilities. The effect of measurement errors and the stability of individual PD control are analyzed in this paper. The central result is a projection method for the elimination of contradicting control commands, which is applicable to general decentralized control schemes. Experimental results are presented for a 2RRR/RR implementation. The results confirm that the proposed method reduces antagonistic control forces up to measurement errors and model uncertainties.

While redundantly actuated PKM (RA-PKM) possess desirable kinematic and dynamic properties, their control is impeded by the occurrence of antagonistic control forces. Such forces are observed in non-linear model-based as well as in decentralized control schemes. In this paper it is outlined that such antagonistic forces are inherent to the decentralized control of RA-PKM, and also that calibration errors and finite encoder resolutions cause antagonistic control forces and excited vibrations, hence energy loss and instabilities. The effect of measurement errors and the stability of individual PD and computed torque control is analyzed. They are shown to yield asymptotically stable setpoint control. A central result of this paper is a projection method for the elimination of contradicting control commands. This method is valid independently of the actual control scheme. Experimental results are presented for a 2RRR/RR implementation. The results confirm that the proposed scheme reduces antagonistic control forces up to measurement errors and model uncertainties.

The analysis of singularities is a central aspect in the design of robotic manipulators. Such analyses are usually based on the use of geometric parameters like DH parameters. However, the manipulator kinematics is naturally described using the concept of screws and twists, associated to Lie groups and algebras. These give rise to general and coordinate-invariant singularity conditions on the manipulator geometry. In this setting no restrictions are imposed onto the type of joints, as it is the case when using DH parameters. In this paper a single closed-form equation is presented that gives a complete description of the singularity locus of an arbitrary regional manipulator in terms of two joint variables and all design parameters, expressed by joint screw coordinates, together with the coordinates for the wrist centre. Some examples are reported, and it is shown that the expression can be used to analyse bifurcations in the singularity locus. The simple form of the condition should make it useful for practical design as well as for a deeper understanding of singularities.

Actuation redundancy increases and homogenizes the kinematic dexterity and stiffness as well as the force distribution among the actuators of parallel-kinematics machines (PKMs). It also allows for internal prestresses within the PKM without affecting the environment that can potentially be used to account for secondary tasks, such as active stiffness and backlash-avoiding control. However, in the presence of geometric uncertainties, this feature can become a serious problem, since then, control forces may be annihilated, or even some of the intentional prestress components may interfere with the environment. While model uncertainties can generally be tackled with robust-control concepts, actuation redundancy of PKM impedes the use of established robust-control schemes. The effect of such uncertainties and the applicability of standard model-based control schemes are analyzed in this paper. It is shown that geometric uncertainties lead to parasitic perturbation forces that cannot be compensated by adjustment of the controls. An amended version of the augmented PD and computed torque-control scheme is proposed that does not suffer from such effects.

Generic manipulators possess the desireable properties that their set of singulartities is a smooth manifold, and that the drop of rank of the manipulator Jacobian is bounded. A sufficient condition for genericity is the transverse-regularity of its Jacobian mapping in any configuration. In this paper a necessary and sufficient condition for transverse-regularity is presented. The condition is based on the manipulator's joint screws and their screw products. It is also shown that a manipulator is non-generic if it can attain a pose where the rank of the manipulator's screw system together with the screw products is not the maximal rank of the Jacobian.

A closed expression for repeated partial derivatives of the inverse mass matrix is developed. It rests on the factorization of the inverse mass matrix in terms of a single configuration-dependent block-diagonal matrix. Thereupon, partial derivatives of the inverse mass matrix are given in terms of derivatives of the diagonal blocks. It turns out that the derivatives are nonvanishing only for a single block of this block-diagonal matrix. The result for open kinematic chains is extended to the mass matrix of mechanisms with kinematic loops

In this paper redundant actuation is used to generate internal preload that would not interfere with the task. This preload is controlled in order to achieve a desired end-effector stiffness, i.e. a desired relationship of applied forces and resulting displacements. This active stiffness yields immediate counteractions to load variations and thus strengthens the integrity of the manipulators. Differential EE-stiffness is defined and a stiffness control scheme is proposed. Results are shown for a planar manipulator

Redundant actuation of parallel manipulators admits internal forces without generating end-effector forces (preload). Preload can be controlled in order to prevent backlash during the manipulator motion. Such controlis based on the inverse dynamics. The general solution of the inverse dynamics of redundantly actuated parallel manipulators is given. For the special case of simple over actuation an explicit solution is derived in terms of a single preload parameter. With this formulation a computational effcient open-loop preload control is developed and applied to the elimination of backlash. Its simplicity makes it applicable in real-time applications. The approach is exemplified for a planar 4RRR manipulator.

Redundant actuation of parallel manipulators can lead to internal forces without generating end-effector forces (preload). Preload can be controlled in order to prevent backlash during the manipulator motion. Such control is based on the inverse dynamics. The general solution of the inverse dynamics of redundantly actuated parallel manipulators is given. For the special case of simple overactuation an explicit solution is derived in terms of a single preload parameter. With this formulation a computational efficient open-loop preload control is developed and applied to the elimination of backlash. Its simplicity makes it applicable in real-time applications. Results are given for a planar 4RRR manipulator and a spatial heptapod.

An iterative solution method for the inverse kinematics (IK) of redundant serial manipulators (SM) is proposed that circumvents collision of manipulator and obstacles. Artificial potential fields are assigned to possibly colliding objects. A predictor-perturbation-corrector (PPC) algorithm accomplishes the IK while the manipulator's end-effector (EE) is tracing a prescribed path. The predictor step achieves geometric tracking of the target EE configuration. An intermediate perturbation step adjusts the manipulator posture away from obstacles. A succeeding corrector step amends the perturbed configuration in accordance with the target EE configuration. The gradients of the artificial potential fields that are tangential to the self motion manifold of the manipulator are considered for perturbation. At each time step the three PPC steps are applied iteratively. The number of necessary PPC steps depends on the required distance to the obstacles. The algorithm is applicable for off-line planing and in real time implementations. It can be extended straight forward to parallel manipulators. Special attention is given to object modelling using artificial potentials functions. Results are shown for a planar 5R and a spatial 10R manipulator.