Robot Kinematics and Dynamics Explained

This article is the first in the “Intermediate Level” series for those wondering “How do you build a robot?“
If you’re just getting started with robot design:

Robot kinematics and dynamics are the disciplines that define the mathematical relationship between a robot’s joint movements and its end-effector position, and they form the foundation of robotic control systems. In this article, we cover the kinematic modeling methods commonly used in robot arms, the approaches for deriving dynamic equations, and the kinematic constraints specific to mobile robots.

Forward Kinematics: From Joint Angles to End-Effector Position

Forward kinematics is the problem of calculating the end-effector’s position and orientation in space given the robot’s known joint angles. In a multi-joint robot arm, this calculation is carried out through the sequential multiplication of homogeneous transformation matrices that define the transformation between each joint. Denavit-Hartenberg (DH) parameters are widely used in robotics to systematically construct these matrices. DH notation defines the geometric relationship between each pair of joints using four parameters: link length, link twist, joint offset, and joint angle. Tabulating these four parameters for every joint produces a standard mathematical model of the robot’s entire kinematic chain, which can then be directly converted into matrix multiplication in software.

Inverse Kinematics: From Target Position to Joint Angles

Inverse kinematics works in the opposite direction from forward kinematics: given a desired target position and orientation for the end-effector, it calculates the joint angles that will achieve that target. This problem is mathematically far more complex than forward kinematics, since there is often more than one set of solutions (for example, reaching a target with an “elbow-up” or “elbow-down” configuration) or no solution at all. In simple robot geometries, inverse kinematics equations can be solved analytically (using trigonometric relationships), while numerical (iterative) methods are generally preferred for complex robot arms with six or more axes. One commonly used numerical approach is the Jacobian-based iterative solution, which uses the inverse of the Jacobian matrix to gradually approach the target.

The Jacobian Matrix: Velocity Kinematics and Singularities

The Jacobian matrix is a linear transformation that defines how small changes in joint angles correspond to changes in end-effector velocity. This matrix plays a critical role not only in velocity kinematics, but also in identifying singularity conditions the robot may encounter in certain configurations. When a robot reaches a configuration where the rank of the Jacobian matrix drops, it loses the ability to move in certain directions; in practice, this can cause the robot to experience sudden and uncontrolled joint velocities in that region. For this reason, keeping the planned path as far as possible from singularity regions is an important design criterion in robot trajectory planning.

Dynamic Modeling: The Newton-Euler and Lagrange Approaches

While kinematic modeling deals only with position and velocity relationships, dynamic modeling defines the force and torque relationships that produce the robot’s motion. Two fundamental approaches are widely used in robot dynamics. The Newton-Euler method sets up separate force and moment balance equations for each link, performing a forward velocity-acceleration calculation from the base to the tip, followed by a backward force-moment calculation from the tip to the base. Because this method is computationally efficient, it is preferred in real-time control applications. The Lagrange method, by contrast, derives the equations of motion from the system’s total kinetic and potential energy; since this approach offers a more mathematically systematic structure, it is widely used in the analytical study of robot dynamics and the theoretical development of control algorithms. Since both methods describe the same physical system, they theoretically produce equivalent results, though their computational complexity and derivation processes differ.

Mobile Robot Kinematics: Holonomic and Nonholonomic Constraints

Unlike fixed-base robot arms, the kinematic model of a wheeled mobile robot must also account for the motion constraints at the point where the wheels contact the ground. In a differential-drive robot, the fact that the robot can instantaneously move only forward-backward and turn, but cannot slide sideways around its own axis, is an example of a nonholonomic constraint; this constraint means that the robot’s degrees of freedom in its configuration space exceed the number of controllable inputs. In contrast, robots with mecanum wheels or omni wheels are classified as holonomic systems, since their wheel designs allow them to move instantly in any direction. Robots with Ackermann steering systems (car-like steering geometry) are subject to a separate set of nonholonomic constraints that require the inner and outer wheels to turn at different angles during a turn. These kinematic differences directly affect which trajectory planning algorithm should be chosen for a mobile robot.

Modeling Software and Simulation Environments

Today, robot kinematics and dynamics are generally modeled using software tools rather than calculated by hand. MATLAB’s Robotics System Toolbox can automatically generate forward and inverse kinematics calculations from DH parameters, while open-source Python libraries (such as PyBullet and modern_robotics) are widely used for dynamic simulation and control algorithm development. These tools speed up the design process by allowing kinematic and dynamic behavior to be verified before a physical prototype is built.

Conclusion

Robot kinematics and dynamics form the foundation for making a robot’s motion mathematically predictable and controllable. Forward and inverse kinematics calculations, velocity relationships via the Jacobian matrix, dynamic equations via the Newton-Euler and Lagrange methods, and kinematic constraints specific to mobile robots are the fundamental mathematical tools encountered in an intermediate-level robotics project. In the next installment of the series, we will cover how these models are combined with noisy real-world sensor data — that is, sensor fusion and filtering techniques.

Frequently Asked Questions

What is the difference between forward kinematics and inverse kinematics? Forward kinematics calculates the end-effector’s position from known joint angles, while inverse kinematics calculates the joint angles required to reach a targeted position.

What are Denavit-Hartenberg (DH) parameters? DH parameters are a notation method that defines the geometric relationship between a robot’s joints using four standard parameters (link length, twist, offset, angle) and systematizes forward kinematics calculation.

Why is the Jacobian matrix important? The Jacobian matrix defines the relationship between joint velocities and end-effector velocity, and it allows singularity configurations, where the robot loses its ability to move, to be identified.

What is the difference between the Newton-Euler and Lagrange methods? The Newton-Euler method produces computationally efficient results by setting up force-moment balances link by link, while the Lagrange method follows a more systematic but computationally heavier derivation process based on the system’s energy.

What is the difference between holonomic and nonholonomic robots? Holonomic robots (such as those with mecanum wheels) can move instantly in any direction, while nonholonomic robots (such as differential-drive robots) cannot move directly in certain directions (such as sideways).