Inverse Kinematics Formulation Research Articles

Static Shape Control of Soft Continuum Robots Using Deep Visual Inverse Kinematic Models

Soft continuum robots are highly flexible and adaptable, making them ideal for unstructured environments such as the human body and agriculture. However, their high compliance and maneuverability make them difficult to model, sense, and control. Current control strategies focus on Cartesian space control of the end-effector, but few works have explored full-body control. This study presents a novel image-based deep learning approach for closed-loop kinematic shape control of soft continuum robots. The method combines a local inverse kinematics formulation in the image space with deep convolutional neural networks for accurate shape control that is robust to feedback noise and mechanical changes in the continuum arm. The shape controller is fast and straightforward to implement; it takes only a few hours to generate training data, train the network, and deploy, requiring only a web camera for feedback. This method offers an intuitive and user-friendly way to control the robot's 3-D shape and configuration through teleoperation using only 2-D hand-drawn images of the desired target state without the need for further user instruction or consideration of the robot's kinematics.

A Compliant Manipulator for Confined Space Tissue Diagnostics: Kinematic and Force Analyses and Initial Characterization Experiments

Abstract Minimally invasive procedures employ continuum manipulators, however internal human anatomy presents challenges relating to size, dexterity, and workspace for these manipulators. This manuscript presents modeling, kinematic analysis, prototyping, and characterization of a micro-robotic manipulator for transurethral palpation of bladder tissue. The proposed micro-robot consists of two subsystems: a tendon-driven continuum segment with an elastic tube encompassing each joint for compliance and structural integrity, and a hyper-spherical joint ensuring higher dexterity and manipulability with a comprehensive actuation and modeling approach. The forward kinematics follow the Denavit-Hartenberg formulation. A developed differential Jacobian inverse kinematics formulation prevents motion singularities for desired poses while operating in the confined space. The simulated kinematic results confirm the dexterity and reach of the proposed micro-robot. A strain energy quasi-static model is developed for a single continuum module. The model is evaluated for tension-bend angle relationships as function of tube material and geometry, and joint length. Limited functionality continuum modules (4mm outside diameter) with four different joint lengths, (3, 6, 9, 12) mm, are prototyped for tension-bend angle characterization using a computer vision outfitted experimental setup. An equivalent shear modulus relationship for the elastic tube for selected joint length values and bend angles is developed using experimental results. The tension-bend angle response is nonlinear and function of tube properties, joint geometry, and their interactions. The comparison of the experimental and quasi-static model results shows high fidelity for use in predicting the robot continuum segment behavior.

A framework for practically effective creation of postprocessors for 5-axis CNC machines with all possible configurations and working mechanisms

The 5-axis CNC (Computer Numerical Control) machining plays an important role in manufacturing, especially in making parts of complex shapes such as turbine or propeller blades. However, there are always challenges of generating 5-axis CNC machining programs when working with different 5-axis CNC machines of various structures; and 5-axis CNC programs must be automatically generated by CNC postprocessors which calculate the inverse kinematics and convert the Cutter Location data to the CNC programs that are used for operating 5-axis CNC machines. Since the family of 5-axis CNC machines has a wide spectrum of machine configurations, with hundreds of mechanisms of 5 degrees of freedom, it is therefore practically challenging for engineers and CNC machine operators to create CNC postprocessors for specific 5-axis CNC machines. It is more challenging to create CNC postprocessors if engineers or CNC operators do not have strong backgrounds and professional skills in mathematical modeling and kinematics of machines. This paper presents a universal and intuitive framework and practical guidance to create CNC postprocessors for all 5-axis CNC machines, with the focus on a novel mathematical formulation of inverse kinematics of three main groups of 5-axis CNC machines. The case studies of creating CNC postprocessors with the commercially available 5-axis CNC machines were successfully demonstrated, in which the simulation scenarios and experiments were implemented to verify the created CNC postprocessors. The proposed frameworks and guides for generating CNC postprocessors can be conveniently and effectively applied in industrial practices, without the required strong backgrounds and skills for engineers or CNC operators in mathematical modeling and kinematics of machines, especially mathematical modeling of multibody systems.

Machine Learning Sequential Methodology for Robot Inverse Kinematic Modelling

The application of robots is growing in most countries, occupying a relevant place in everyday environments. Robots are still affected by errors due to their limitations, which may compromise the final performance. Accurate trajectories and positionings are strict requirements that robots have to satisfy and may be studied by the inverse kinematic (IK) formulation. The IK conventional numerical techniques are computationally intensive procedures, focusing on the robot joint values simultaneously and increasing the complexity of the solution identification. In this scenario, Machine Learning strategies may be adopted to achieve effective and robust manipulator’s IK formulation due to their computational efficiency and learning ability. This work proposes a machine learning (ML) sequential methodology for robot inverse kinematics modeling, iterating the model prediction at each joint. The method implements an automatic Denavit-Hartenberg (D-H) parameters formulation code to obtain the forward kinematic (FK) equations required to produce the robot dataset. Moreover, the artificial neural network (ANN) architecture is selected as a structure and the number of layers in combination with the hidden neurons per layer is defined by an offline optimization phase based on the genetic algorithm (GA) technique for each joint. The ANN is implemented with the following settings: scaled conjugate gradient as training function and the mean squared error as the loss function. Different network architectures are examined to validate the IK procedure, ranging from global to sequential and considering the computation direction (from end-effector or from basement). The method is validated in the simulated and experimental laboratory environment, considering articulated robots. The sequential method exhibits a reduction of the mean squared error index of 42.7–56.7%, compared to the global scheme. Results show the outstanding performance of the IK model in robot joint space prediction, with a residual mean absolute error of 0.370–0.699 mm in trajectory following 150.0–200.0 mm paths applied to a real robot.

Redundant Manipulator Kinematics and Dynamics on Differentiable Manifolds

Abstract A recently published treatment of nonredundant manipulator kinematics and dynamics on differentiable manifolds is extended to kinematically redundant manipulators. Analysis at the configuration level shows that forward kinematics and dynamics of redundant manipulators are identical to that for nonredundant manipulators. The manifold-based inverse kinematics formulation that is presented for redundant manipulators, in contrast, yields parameterizations of set-valued inverse kinematic mappings at the configuration level, where sharper results are obtained than those presented in the literature using velocity formulations. Explicit expressions are derived for set-valued inverse kinematic mappings for both serial and nonserial (called compound) kinematically redundant manipulators, as functions of vectors of arbitrary parameters. Parameterizations are presented for both manipulator regular configuration manifolds and self-motion manifolds, the latter comprised of sets of inputs that map into the same output. It is shown that kinematically redundant configuration manifolds and self-motion differentiable manifolds are distinctly different and play complementary roles in redundant manipulator kinematics. Computational methods are presented for evaluation of set-valued inverse kinematic mappings, without problem-dependent ad hoc analytical manipulations. Redundant serial and compound manipulator examples are presented to illustrate computation of set-valued inverse kinematic mappings and the use of self-motion manifold mappings in obstacle avoidance applications. Differentiation of configuration level inverse mappings yields inverse velocity and acceleration mappings as functions of time-dependent arbitrary parameters that play a central role in manipulator dynamics and control.

Inverse Kinematics and Dexterous Workspace Formulation for 2-Segment Continuum Robots With Inextensible Segments

The inverse kinematics (IK) problem of continuum robots has been investigated in depth in the past decades. Under the constant-curvature bending assumption, closed-form IK solution has been obtained for continuum robots with variable segment lengths. Attempting to close the gap towards a complete solution, this paper presents an efficient solution for the IK problem of 2-segment continuum robots with one or two inextensible segments (a.k.a, constant segment lengths). Via representing the robots shape as piecewise line segments, the configuration variables are separated from the IK formulation such that solving a one-variable nonlinear equation leads to the solution of the entire IK problem. Furthermore, an in-depth investigation of the boundaries of the dexterous workspace of the end effector caused by the configuration variables limits as well as the angular velocity singularities of the continuum robots was established. This dexterous workspace formulation, which is derived for the first time to the best of the authors knowledge, is particularly useful to find the closest orientation to a target pose when the target orientation is out of the dexterous workspace. In the comparative simulation studies between the proposed method and the Jacobian-based IK method involving 500,000 cases, the proposed variable separation method solved 100% of the IK problems with much higher computational efficiency.

Manipulator Kinematics and Dynamics on Differentiable Manifolds: Part II Dynamics

Abstract The manipulator differentiable manifold kinematics formulation presented in Part I of this paper is employed for forward and inverse kinematic analysis and formulation of ordinary differential equations of motion on disjoint, maximal, singularity free, path-connected components of regular configuration space, for a broad spectrum of manipulators. Existence of forward and inverse configuration mappings throughout maximal singularity free manifold components yields analytical forward and inverse velocity mappings. Efficient computational algorithms for forward and inverse configuration and velocity analysis on a time grid are derived for each of three manipulator categories. Manifold parameterizations are used to transform variational equations of motion in manipulator coordinates to second-order ordinary differential equations of manipulator dynamics, with both input and output coordinates as state variables, eliminating the need for ad hoc derivation of equations of motion. Criteria that define manipulator differentiable manifolds are shown to guarantee that the equations of motion derived are well-posed on maximal singularity free components of manipulator configuration space. This process is illustrated by presenting terms required for the evaluation of equations of motion for three model manipulators. It is shown that computation involved in the evaluation of equations of manipulator kinematics and dynamics can be carried out in real-time on modern microprocessors, supporting the in-line implementation of modern methods of manipulator control.

A Versatile Inverse Kinematics Formulation for Retargeting Motions Onto Robots With Kinematic Loops

Robots with kinematic loops are known to have superior mechanical performance. However, due to these loops, their modeling and control is challenging, and prevents a more widespread use. In this letter, we describe a versatile Inverse Kinematics (IK) formulation for the retargeting of expressive motions onto mechanical systems with loops. We support the precise control of the position and orientation of several end-effectors, and the Center of Mass (CoM) of slowly walking robots. Our formulation safeguards against a disassembly when IK targets are moved outside the workspace of the robot, and we introduce a regularizer that smoothly circumvents kinematic singularities where velocities go to infinity. With several validation examples and three physical robots, we demonstrate the versatility and efficacy of our IK on overactuated systems with loops, and for the retargeting of an expressive motion onto a bipedal robot.

Accuracy Analysis of Machining Trajectory Contemplating Workpiece Dislocation on a Six Degree of Freedom Machining Bed

Machining is material removal from a workpiece. Current spindle technologies allow the material to be removed very quickly but unfortunately this compromises the accuracy of the desired machined trajectory on the workpiece. Proposed solution to the problem is restricting motion of the tool and giving rotation to the workpiece. This paper presents analysis of the accuracy of trajectory of material removal from a workpiece, such that the workpiece rotates with six degrees of freedom, in the presence of error generated due to an offset of the workpiece from centre of the moving platform of the machining bed. The kinematics of the machining bed is, therefore, modeled using as inverse kinematic formulation applying geometric and vector addition method. The mechanism outputs three rotational and three translational motions. The leg length for each of six legs of the bed is computed individually. Moreover, workpiece position offset error is modelled to find actual leg lengths of the bed. Finally accuracy computation model is proposed to analyse the accuracy of the final trajectory of the motion of the workpiece. The models are verified in simulation for a trajectory and validated experimentally on a six degree of freedom (6DOF) machining bed. The results reveal maximum inaccuracy in machining trajectory to be 1% in experiments while less than 1% in simulation. This verifies quality of the mechanism and efficacy of the proposed 6DOF machining bed in machining accuracy.

Kinematic and dynamic analysis of the Gantry-Tau, a 3-DoF translational parallel manipulator

• A closed-from mathematical model for dynamics of the Gantry-Tau is derived. • Kinematics of the Gantry-Tau is formulated and singular conditions are discussed. • Simulation studies are carried out to assess the performance of the proposed models. This paper deals with the mathematical modeling of kinematics and dynamics of the 3-degrees-of-freedom Gantry-Tau manipulator. Compared to many other parallel robots, Gantry-Tau offers a large accessible workspace and high stiffness. The kinematics of Gantry-Tau is presented which includes inverse kinematics formulation for the position, velocity and acceleration of the mechanism. Also, based on the obtained Jacobin matrices, singular configurations of the robot are studied. Afterwards, the equations of the inverse dynamic model of the Gantry-Tau are obtained through two different methods, i.e., virtual work and Newton–Euler. Finally, a case study is performed to verify the correctness of the derived models and investigate their computational efficiency.

Singularity-Tolerant Inverse Kinematics for Bipedal Robots: An Efficient Use of Computational Power to Reduce Energy Consumption

We propose a nonlinear inverse kinematics formulation which solves for positions directly. Compared to various other popular methods that integrate velocities, this formulation can better handle fast, asymmetric, and singular-postured balancing tasks for humanoid robots. We also introduce joint position and velocity boundaries as inequality constraints in the optimization to ensure feasibility. Such boundaries provide safety when approaching or getting away from joint limits or singularities. Besides, mixing positions and velocities in our proposed algorithm facilitates recovery from singularities, which is very difficult for conventional inverse kinematics methods. Extensive demonstrations on the real robot prove the applicability of the proposed algorithm while improving power consumption. Our formulation automatically handles different numerical and behavioral difficulties rising from singularities, which makes it a reliable low-level conversion block for different Cartesian planners.

Kinematic and three-dimensional dynamic modeling of a biped robot

This article focuses on inverse kinematic formulation and dynamic modeling of the Nao biped robot's lower body, accompanied by verification with the joints' angles as experimental data. Dynamic modeling in two different planes is discussed and joint angles for the given positions, nominal conditions, and trajectory computations are simulated and graphically illustrated. A new approach for development of the inverse dynamics on the aforementioned robot's lower body is proposed in this paper, analytically studied, and compared with MSC Adams for two various scenarios of fixed supporting leg and ground contact implementation.

A Polynomial Formulation of Inverse Kinematics of Rolling Contact

Rolling contact has been used by robotic devices to drive between configurations. The degrees of freedom (DOFs) of rolling contact pairs can be one, two, or three, depending on the geometry of the objects. This paper aimed to derive three kinematic inputs required for the moving object to follow a trajectory described by its velocity profile when the moving object has three rotational DOFs and thus can rotate about any axis through the contact point with respect to the fixed object. We obtained three contact equations in the form of a system of three nonlinear algebraic equations by applying the curvature theory in differential geometry and simplified the three nonlinear algebraic equations to a univariate polynomial of degree six. Differing from the existing solution that requires solving a system of nonlinear ordinary differential equations, this polynomial is suitable for fast and accurate numerical root approximations. The contact equations further revealed the two essential parts of the spin velocity: The induced spin velocity governed by the geometry and the compensatory spin velocity provided externally to realize the desired spin velocity.

SU‐F‐BRE‐05: Development and Evaluation of a Real‐Time Robotic 6D Quality Assurance Phantom

Purpose:A 6 degree‐of‐freedom robotic phantom capable of reproducing dynamic tumor motion in 6D was designed to more effectively match solid tumor movements throughout pre‐treatment scanning and radiation therapy. With the abundance of optical and x‐ray 6D real‐time tumor tracking methodologies clinically available, and the substantial dosimetric consequences of failing to consider tumor rotation as well as translation, this work presents the development and evaluation of a 6D instrument with the facility to improve quality assurance.Methods:An in‐house designed and built 6D robotic motion phantom was constructed following the so‐called Stewart‐Gough parallel kinematics platform archetype. The device was then controlled using an inverse kinematics formulation, and precise movements in all six degrees of freedom (X, Y, Z, pitch, roll, and yaw) as well as previously obtained cranial motion, were effectively executed. The robotic phantom movements were verified using a 15 fps 6D infrared marker tracking system (Polaris, NDI), and quantitatively compared to the input trajectory. Thus, the accuracy and repeatability of 6D motion was investigated and the phantom performance was characterized.Results:Evaluation of the 6D platform demonstrated translational RMSE values of 0.196 mm, 0.260 mm, and 0.101 mm over 20 mm in X and Y and 10 mm in Z, respectively, and rotational RMSE values of 0.068 degrees, 0.0611 degrees, and 0.095 degrees over 10 degrees of pitch, roll, and yaw, respectively. The robotic stage also effectively performed controlled 6D motions, as well as reproduced cranial trajectories over 15 minutes, with a maximal RMSE of 0.044 mm translationally and 0.036 degrees rotationally.Conclusion:This 6D robotic phantom has proven to be accurate under clinical standards and capable of reproducing tumor motion in 6D. Consequently, such a robotics device has the potential to serve as a more effective system for IGRT QA that involves both translational and rotational dimensions.Research was partially funded by NIH Grant T32 EB002103‐21 from NIBIB. Contents are solely the responsibility of the authors and do not necessarily represent the official views of the NIBIB or NIH.

Dependence of helicopter pilots’ biodynamic feedthrough on upper limbs’ muscular activation patterns

The involuntary interaction of pilots with vehicles is often an undesired consequence of the biodynamic feedthrough of cockpit vibrations into the control system in relation with the characteristics of the man–machine interface. This work presents a numerical study of how the estimated muscular activation patterns associated with performing basic helicopter piloting tasks may affect the variability of the pilot’s biodynamic feedthrough and admittance. The limbs’ motion is predicted using an inverse kinematics formulation for redundant manipulators imposing the motion of the hand from measurements. Articulation torques are then estimated by inverse dynamics. Activation of the involved muscles is estimated according to the ‘total activation’ paradigm. Equivalent pilot feedthrough is obtained by consistent linearization of the constitutive model of the muscles about the reference activation. The effect on equivalent feedthrough of non-optimal activation, resulting from the addition of torque-less activation modes to the optimal activation, is evaluated and discussed. The multibody model of the pilot’s biodynamic feedthrough is incorporated in a detailed multibody model of a helicopter, to perform coupled bioaeroservoelastic simulations.

A simulation framework for evolution on uneven terrains for synchronous drive robot

This paper presents a simulation framework for evolution on uneven terrains for a wheeled mobile robot (WMR) such as a synchronous drive robot. The framework lends itself as a tool capable of solving various problems, such as forward kinematic-based evolution, inverse kinematic-based evolution, path planning and trajectory tracking. This framework becomes particularly useful when we understand that the evolution problem (and hence, the various associated problems based on evolution) is particularly challenging on uneven terrain. Specifically, it is entailed to bring in the contact constraints posed by the interaction of the wheel and the ground as well as the holonomic constraints as the problem is formulated in a Differential Algebraic Equation setting. The problem becomes all the more crucial as vehicles moving on uneven terrain are becoming the order of the day. Nonetheless, there has not been much literature that deals in length the various aspects that go into the framework. This paper elaborates on the various aspects of the framework, presents simulation results on uneven terrain, where the vehicle evolves without slipping, and also presents substantial quantitative analysis in regard to wheel slippage. The main contributions of this paper are the motion planning using forward kinematic framework and a new formulation of inverse kinematics for wheeled robots on uneven terrains.

A Comparative Study of Three Inverse Kinematic Methods of Serial Industrial Robot Manipulators in the Screw Theory Framework

In this paper, we compare three inverse kinematic formulation methods for the serial industrial robot manipulators. All formulation methods are based on screw theory. Screw theory is an effective way to establish a global description of rigid body and avoids singularities due to the use of the local coordinates. In these three formulation methods, the first one is based on quaternion algebra, the second one is based on dual-quaternions, and the last one that is called exponential mapping method is based on matrix algebra. Compared with the matrix algebra, quaternion algebra based solutions are more computationally efficient and they need less storage area. The method which is based on dual-quaternion gives the most compact and computationally efficient solution. Paden-Kahan sub-problems are used to derive inverse kinematic solutions. 6-DOF industrial robot manipulator's forward and inverse kinematic equations are derived using these formulation methods. Simulation and experimental results are given.

Dynamic Dimensional Synthesis of Delta Robot

The dynamic dimensional synthesis of Delta robot widely used for high-speed pick-and-place operations is studied. On the basis of the inverse kinematic and rigid body dynamic formulation, a novel global dynamic performance index is proposed for minimizing the maximum driving torque of a single active joint. The proposed index can explicitly be expressed in terms of dimensional and inertial parameters as well as system configurations, allowing the singular configurations to be easily found. Various geometrical and performance constraints are taken into account in terms of workspace/machine volume ratio, lower and upper bounds of mapping characteristics in terms of velocity, accuracy and actuation rigidity between the joint space and the operation space. The effects of the performance constraints on the feasible domain of design variables and the performance index are investigated in depth via an example, enabling a set of optimized dimensional parameters to be obtained for achieving a good dynamic performance throughout the entire workspace.

A Two-Level Strategy for Optimizing the Reliability of Redundant Inverse Kinematic Solutions

This paper presents a novel computational algorithm for selecting an optimal inverse kinematic solution in response to a pose specification in a redundant robotic manipulator. In a previous work [9], the optimization of an entropy function was suggested as the selection criteria for determining a solution to the redundant inverse kinematics problem; it was shown that judicious selection of joint values that minimize this entropy function will result in a highly-reliable solution to the inverse kinematics problem. The technique was demonstrated in a simple case study, where a closed-form expression could be developed to describe the pose uncertainty of the manipulator. In this paper, a novel computational technique is presented for minimizing the entropy function which described the positioning uncertainty of the robotic manipulator. This two-level algorithm allows the reliability-based inverse kinematic formulation to be extended to more complex manipulators, where simulation techniques must be used to determine the pose uncertainty of the manipulator. This technical note presents a review of the reliability-based inverse kinematic formulation, a description of the novel computational algorithm used to minimize the entropy function describing the reliability of the inverse kinematic solution, and an example implementation in a four degree of freedom manipulator.