What kinematic equations are important for industrial robots?

A kinematic model is a mathematical representation of a robot’s motion that focuses on the geometric relationships between elements and ignores external forces that affect motion. Kinematic models are used for planning and controlling the paths of industrial robots.

Forward kinematics determines the current position and orientation of the end effector based on the angles of each joint in the robot arm. A series of transformation matrices containing joint angles, length, and offset parameters is used for the calculations.

The transformation matrices are multiplied sequentially from the robot base to the end effector. The result is the position in three-dimensional coordinates and the orientation (in terms of angles) of the end effector. Inverse kinematics reverses the process, determining the required joint angles based on the position of the end effector (Figure 1).

Figure 1. Defining robot joint positions using forward or inverse kinematics. (Image: MathWorks)

Denavit-Hartenberg convention

The Denavit-Hartenberg (DH) convention is a standardized way of representing a robot’s geometry and joint angles. It details how to attach coordinate frames to the joints in a robot, defining the relative position and orientation between adjacent joints.

Each joint, or link, is described by four parameters, including the link length (a), joint angle (θ), link offset (d), and joint twist (α) (Figure 2). The four parameters are used to calculate transformation matrices for each joint. The matrices can be multiplied to determine the overall transformation from a robot’s base to the end-effector. Control systems use DH parameters to calculate the required joint angles to achieve desired end-effector poses.

Figure 2. Example of DH parameters. (Image: MDPI sensors)

Jacobian matrix

DH convention can be used together with Jacobian matrices to relate joint velocities to end-effector velocities. Joint velocities can describe how fast the joint is rotating for a revolute joint and how fast the joint is extending or contracting for a prismatic joint.

The Jacobian matrix (J) is a matrix of partial derivatives that describes the relationship between the joint velocities and the velocity of the end-effector. In robot control systems, J is used to map joint velocities to the end-effector’s velocity.

Transposing J allows calculation of joint torques needed to produce desired forces at the end-effector. In motion planning, J is used to identify and avoid singularities where the robot loses the ability to move freely and precisely. A singularity occurs when the determinant of J becomes zero.

Singularities

In a singularity, the mathematical relationship between joint angles and end-effector position breaks down, and the robot loses one or more degrees of freedom. Singularities occur when multiple joints align in a way that prevents the robot from moving in one or more directions. That can result in loss of control, instability or jerky movement. Singularity configurations can be avoided with careful design of motion trajectories.

Manipulability

Singularities are one form of limited manipulability. Manipulability measures how well the robot can change the end effector position based on the current joint configuration. Manipulability is measured using the singular value decomposition of J to compute the manipulability ellipsoid.

The manipulability ellipsoid visually represents the robot’s ability to move in different directions. An ellipsoid with a larger volume indicates higher manipulability. A ratio of the ellipsoid axis close to 1 indicates equal manipulability in all directions, known as isotropic manipulability. A large ratio suggests a singularity is approaching.

Summary

Kinematic equations are used to determine the optimal trajectory for moving from one pose to another and to avoid difficulties like singularities that can limit a robot’s motion. The DH convention is a standardized way of representing a robot’s geometry and joint angles. The Jacobian matrix provides a computational framework for optimizing robot movements.