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Robot kinematics
Robot kinematics
Robot kinematics
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Robot kinematics applies geometry to the study of the movement of multi-degree of freedom kinematic chains that form the structure of robotic systems.[1][2] The emphasis on geometry means that the links of the robot are modeled as rigid bodies and its joints are assumed to provide pure rotation or translation.

Robot kinematics studies the relationship between the dimensions and connectivity of kinematic chains and the position, velocity and acceleration of each of the links in the robotic system, in order to plan and control movement and to compute actuator forces and torques. The relationship between mass and inertia properties, motion, and the associated forces and torques is studied as part of robot dynamics.

Kinematic equations

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Further information: Kinematic equation

A fundamental tool in robot kinematics is the kinematics equations of the kinematic chains that form the robot. These non-linear equations are used to map the joint parameters to the configuration of the robot system. Kinematics equations are also used in biomechanics of the skeleton and computer animation of articulated characters.

Forward kinematics uses the kinematic equations of a robot to compute the position of the end-effector from specified values for the joint parameters.[3] The reverse process that computes the joint parameters that achieve a specified position of the end-effector is known as inverse kinematics. The dimensions of the robot and its kinematics equations define the volume of space reachable by the robot, known as its workspace.

There are two broad classes of robots and associated kinematics equations: serial manipulators and parallel manipulators. Other types of systems with specialized kinematics equations are air, land, and submersible mobile robots, hyper-redundant, or snake, robots and humanoid robots.

Forward kinematics

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Main article: Forward kinematics

Forward kinematics specifies the joint parameters and computes the configuration of the chain. For serial manipulators this is achieved by direct substitution of the joint parameters into the forward kinematics equations for the serial chain. For parallel manipulators substitution of the joint parameters into the kinematics equations requires solution of the a set of polynomial constraints to determine the set of possible end-effector locations.

Inverse kinematics

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Main article: Inverse kinematics

Inverse kinematics specifies the end-effector location and computes the associated joint angles. For serial manipulators this requires solution of a set of polynomials obtained from the kinematics equations and yields multiple configurations for the chain. The case of a general 6R serial manipulator (a serial chain with six revolute joints) yields sixteen different inverse kinematics solutions, which are solutions of a sixteenth degree polynomial. For parallel manipulators, the specification of the end-effector location simplifies the kinematics equations, which yields formulas for the joint parameters.

Robot Jacobian

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The time derivative of the kinematics equations yields the Jacobian of the robot, which relates the joint rates to the linear and angular velocity of the end-effector. The principle of virtual work shows that the Jacobian also provides a relationship between joint torques and the resultant force and torque applied by the end-effector. Singular configurations of the robot are identified by studying its Jacobian.

Velocity kinematics

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The robot Jacobian results in a set of linear equations that relate the joint rates to the six-vector formed from the angular and linear velocity of the end-effector, known as a twist. Specifying the joint rates yields the end-effector twist directly.

The inverse velocity problem seeks the joint rates that provide a specified end-effector twist. This is solved by inverting the Jacobian matrix. It can happen that the robot is in a configuration where the Jacobian does not have an inverse. These are termed singular configurations of the robot.

Static force analysis

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The principle of virtual work yields a set of linear equations that relate the resultant force-torque six vector, called a wrench, that acts on the end-effector to the joint torques of the robot. If the end-effector wrench is known, then a direct calculation yields the joint torques.

The inverse statics problem seeks the end-effector wrench associated with a given set of joint torques, and requires the inverse of the Jacobian matrix. As in the case of inverse velocity analysis, at singular configurations this problem cannot be solved. However, near singularities small actuator torques result in a large end-effector wrench. Thus near singularity configurations robots have large mechanical advantage.

Fields of study

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Robot kinematics also deals with motion planning, singularity avoidance, redundancy, collision avoidance, as well as the kinematic synthesis of robots.[4]

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Robot kinematics

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from Grokipedia
Robot kinematics is the branch of that analyzes the geometric relationships between a robot's parameters—such as angles for revolute joints or displacements for prismatic joints—and the position and orientation of its end-effector or other components in space, without regard to the forces or torques causing the motion. This field focuses on positions, velocities, and higher-order derivatives of poses for rigid bodies connected by , treating motion as a purely geometric phenomenon. It applies to diverse robotic systems, including serial manipulators, parallel mechanisms, and mobile platforms, enabling tasks like precise positioning and trajectory prediction. A core component of robot kinematics is forward kinematics, which maps specified configurations to the corresponding end-effector pose using transformations, often represented by homogeneous matrices. For serial chains, this involves composing individual link transformations recursively. The Denavit-Hartenberg (DH) convention standardizes this process by defining four parameters per —link length (a), link twist (α), joint angle (θ), and joint offset (d)—to parameterize the spatial relationships between adjacent links, facilitating efficient computation for multi-degree-of-freedom systems. This method, introduced in the mid-20th century, remains foundational for modeling manipulators with up to six or more s. In contrast, solves the reverse problem: finding joint variables that achieve a target end-effector pose, which can yield multiple solutions or none, depending on the robot's workspace and redundancy. Velocity kinematics extends this to rates of change, using Jacobians to relate joint velocities to end-effector linear and angular velocities, essential for real-time control and singularity analysis. These concepts underpin applications in industrial automation, surgical robotics, and autonomous vehicles, where accurate kinematic models ensure collision-free motion and task execution.

Basic Concepts

Definition and Scope

Robot kinematics is the study of the motion of robotic systems without considering the forces or torques that produce it, concentrating instead on the positions, velocities, and accelerations of the robot's components relative to one another. This field examines how joint configurations determine the pose of end-effectors or other key points, using geometric and algebraic models to describe spatial relationships. The origins of kinematics trace back to classical mechanics in the 18th century, where Leonhard Euler laid foundational work on the three-dimensional motion of rigid bodies, including systematic analyses of rotations and orientations. These principles were later adapted to during the mid-20th century, particularly with the development of early industrial manipulators like the in the , which necessitated kinematic models for programming and control. serves as a prerequisite for robot dynamics, which incorporates forces, but remains distinct by focusing solely on geometric constraints and motion descriptions. In scope, robot kinematics applies broadly to serial manipulators, parallel mechanisms, mobile platforms such as wheeled , and systems, emphasizing mathematical representations of achievable motions. Central concepts include the configuration space (C-space), an n-dimensional manifold representing all possible joint variable combinations, where n equals the ; the task space, which captures the end-effector's position and orientation in the operational environment; and the workspace, the bounded subset of task space reachable by the end-effector under joint limits. These elements enable analysis of and constraints essential for robot design and operation.

Coordinate Frames and Transformations

In robot kinematics, coordinate frames provide a structured way to describe the position and orientation of robot components relative to one another in . The base frame is typically fixed to the robot's stationary base or the world coordinate system, serving as the reference for all measurements. Intermediate link frames are assigned to each rigid link in the robot's , capturing the local position and orientation of that link relative to the previous one. The end-effector frame is attached to the tool or gripper at the robot's tip, representing the final pose where tasks are performed. These frames enable the of complex robot configurations into manageable parts, facilitating analysis of motion without considering dynamics. Representing orientations within these frames requires methods to describe rotations, as pure translations are insufficient for rigid bodies. Euler angles parameterize rotations as three sequential angles about the x, y, and z axes, offering an intuitive decomposition but suffering from gimbal lock—a singularity where two axes align, losing a degree of freedom and causing discontinuities in representation. Roll-pitch-yaw (RPY) angles, a common Euler angle convention in and , typically involve sequential rotations: first yaw about the z-axis, then pitch about the y-axis, and roll about the x-axis. These are often applied using intrinsic rotations about body-fixed axes, aligning with intuitive motion descriptions, but suffer from gimbal lock near pitch angles of ±90°. Quaternions, four-dimensional vectors (w, x, y, z) with unit norm, avoid singularities and provide smooth for rotations, making them preferable for computational efficiency and avoiding the 3-to-4 redundancy of matrices, despite requiring normalization to prevent drift. These representations convert to 3x3 rotation matrices for use in transformations, ensuring orthogonal preservation of distances. To combine rotation and translation into a single operation, homogeneous transformation matrices are employed, augmenting 3x3 matrices with a 3x1 vector in a 4x4 structure: [Rp01×31]\begin{bmatrix} \mathbf{R} & \mathbf{p} \\ \mathbf{0}_{1 \times 3} & 1 \end{bmatrix} where R\mathbf{R} is the and p\mathbf{p} is the position vector. This form, part of the special Euclidean group SE(3), allows to compose successive transformations, such as transforming a point from one frame to another by premultiplying its homogeneous coordinates [xT,1]T[ \mathbf{x}^T, 1 ]^T. For a robot with n links, the overall transformation from base to end-effector is the chain product T=A1A2⋯AnT = A_1 A_2 \cdots A_n, where each AiA_i is the homogeneous matrix for the i-th link transformation. This chaining underpins forward kinematics computations by propagating poses through the robot structure.

Kinematic Chains

Serial Manipulators

Serial manipulators, also known as serial-link or open-chain robots, form the backbone of most industrial robotic systems due to their straightforward and versatility. These robots consist of a fixed base connected to a series of rigid links joined end-to-end by actuators, typically revolute joints that enable or prismatic joints that allow linear translation. This open structure ensures that motion propagates sequentially from the base to the end-effector without forming closed loops, allowing for a tree-like where each connects a parent link to a child link. The mobility of serial manipulators is determined by the number of independent joint motions, with the (DOF) equaling the count of in non-redundant cases, where each contributes one DOF to the overall . This configuration enables precise control over the end-effector's position and orientation in task , particularly for manipulators with 6 DOF to achieve full spatial dexterity. Modeling such chains often involves establishing coordinate at each to track transformations along the structure. Classic examples of serial manipulators include the PUMA 560, a 6-DOF articulated arm developed for assembly and welding tasks, which exemplifies the anthropomorphic design mimicking human arm motion. Another is the robot, featuring four DOF with two horizontal revolute joints, a vertical , and a rotational , optimized for high-speed pick-and-place operations in electronics manufacturing. These designs highlight the adaptability of serial structures to diverse applications. One key advantage of serial manipulators is their extensive reachable workspace, as the sequential extension of links allows access to a broad volume without mechanical interference. However, this open-chain architecture results in lower structural stiffness, leading to greater end-effector deflection under or dynamic loads compared to closed-loop alternatives.

Parallel and Hybrid Manipulators

Parallel manipulators, also known as closed-chain robots, consist of multiple kinematic chains or legs that connect a fixed base platform to a moving end-effector platform, forming closed loops that impose geometric constraints on the motion. This architecture differs fundamentally from serial manipulators by distributing the end-effector load across several parallel paths, enhancing structural integrity. The seminal example is the Stewart-Gough platform, introduced by V. E. Gough for tire testing in 1947 and later adapted by D. Stewart for flight simulation in 1965, featuring six extensible legs each connected via universal joints to the base and spherical joints to the platform, enabling full six-degree-of-freedom (6-DOF) spatial motion. The mobility, or degrees of freedom (DOF), of parallel manipulators is calculated using the Grübler-Kutzbach criterion, which quantifies the net motion after accounting for joint freedoms and loop constraints: DOF = 6(n - j - 1) + ∑ f_i, where n is the number of links including the fixed frame, j is the number of joints, and f_i is the freedom of the i-th joint; this often yields fewer effective DOF than the number of actuators due to overconstraints, ensuring precise but restricted motion. In practice, such systems achieve high precision in constrained tasks, as the closed loops reduce cumulative errors seen in open serial chains. Representative examples include the , invented by Reymond Clavel in 1985 at EPFL for high-speed pick-and-place operations in industries like and electronics assembly, utilizing three parallelogram linkages driven by three rotary motors to achieve 3-DOF translational motion with exceptional acceleration up to 20 g in experimental setups. Hybrid manipulators combine parallel structures with serial or cable-driven elements to mitigate limitations; for instance, hybrid cable-driven systems employ rigid parallel legs augmented by flexible cables, extending the workspace for applications like large-scale handling while preserving stiffness, as demonstrated in designs for overhead manipulation tasks. Parallel and hybrid manipulators provide advantages such as superior stiffness-to-weight ratios, enabling payloads up to several times their own mass with positioning accuracies below 0.1 mm, and high dynamic speeds due to low from fixed actuators on the base. However, these benefits come at the cost of a limited and complex workspace shaped by leg interferences and singularities, as well as challenges in forward due to multiple assembly modes. In comparison to serial manipulators, parallel designs excel in precision-oriented tasks but sacrifice reach.

Forward Kinematics

Formulation and Denavit-Hartenberg Parameters

The forward kinematics of serial robotic manipulators is formulated by composing homogeneous transformation matrices that map the position and orientation from the base coordinate frame to the end-effector frame through successive link transformations. The Denavit-Hartenberg (DH) convention standardizes this process by assigning a coordinate frame to each link in the , enabling a compact parameterization of the relative pose between frames. Introduced in the seminal work on kinematic notation for mechanisms, this method uses matrix algebra to describe the geometry and configuration of lower-pair joints in serial chains. Each link ii in the chain is characterized by four DH parameters: the link length aia_i, which is the distance along the common normal from the zi−1z_{i-1} axis to the ziz_i axis; the link twist αi\alpha_i, the angle between the zi−1z_{i-1} and ziz_i axes measured in the plane perpendicular to the common normal; the joint angle θi\theta_i, the rotation about the zi−1z_{i-1} axis from the common normal to the xix_i axis; and the link offset did_i, the distance along the ziz_i axis from the common normal to the origin of frame ii. These parameters capture both the fixed geometry ( aia_i and αi\alpha_i ) and the variable configuration ( θi\theta_i for revolute joints or did_i for prismatic joints ). The coordinate frames are assigned systematically: the ziz_i axis aligns with the axis of joint i+1i+1; the xix_i axis lies along the common perpendicular between zi−1z_{i-1} and ziz_i; and the yiy_i axis completes a right-handed orthogonal frame, with the origin at the intersection of xix_i and ziz_i. For the base frame (frame 0), the z0z_0 axis aligns with the first joint axis, and the x0x_0 axis points toward the common normal with z1z_1. This assignment ensures a unique, minimal description for open serial chains, though the common normal is undefined if axes are parallel or intersecting. The transformation from frame i−1i-1 to frame ii, denoted i−1Ai^{i-1}\mathbf{A}_i, is a 4×4 homogeneous matrix derived as the product of basic rotations and translations corresponding to the DH parameters: [cos⁡θi−sin⁡θicos⁡αisin⁡θisin⁡αiaicos⁡θisin⁡θicos⁡θicos⁡αi−cos⁡θisin⁡αiaisin⁡θi0sin⁡αicos⁡αidi0001]\begin{bmatrix} \cos \theta_i & -\sin \theta_i \cos \alpha_i & \sin \theta_i \sin \alpha_i & a_i \cos \theta_i \\ \sin \theta_i & \cos \theta_i \cos \alpha_i & -\cos \theta_i \sin \alpha_i & a_i \sin \theta_i \\ 0 & \sin \alpha_i & \cos \alpha_i & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix}
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