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Kinematics

Definition

Kinematics is the study of the geometry of motion, including positions, orientations, velocities and accelerations, without using the forces and torques that cause the motion. In robotics, it relates joint configurations and movement to the resulting arrangement and motion of the robot's parts.

What it means

A kinematic model describes how connected parts can move according to their geometry and constraints. Link lengths, joint types and reference frames matter. The model can describe both a stationary arrangement and how that arrangement changes over time.

Forward kinematics starts with joint coordinates and computes an end-effector pose. Inverse kinematics starts with a desired pose and seeks joint coordinates that produce it. Velocity kinematics relates joint rates to motion of the end effector, commonly through a Jacobian matrix.

Dynamics asks additional questions about inertia, applied forces and torques. Acceleration can appear in a kinematic description because it is a rate of change of velocity; calculating the forces needed to produce that acceleration is a dynamics question.

How it works

Define the mechanism and attach coordinate frames with explicit units and angle conventions. Write geometric relationships that map joint variables to link positions and orientations. For a serial arm, these relationships combine the transformations contributed by successive joints and links.

Differentiate appropriate position relationships to obtain velocity relationships. Differentiate again when acceleration is needed. For general spatial robots, orientation and angular velocity require suitable representations rather than treating every stored coordinate derivative as interchangeable.

Use the model to answer the specific geometric question, while retaining constraints such as joint limits and closed-loop relationships. A reachable endpoint does not establish a collision-free path, and a geometric trajectory does not establish that available actuators can execute it. These are additional analyses.

Three practical examples

• A robot-arm simulator draws the end-effector position from specified joint angles.
• An animation system solves joint arrangements that place a character's hand at a target location.
• A motion-analysis exercise computes the speed and direction of a point on a rotating link from the link's angular rate.

Think of it this way

Think of describing a bicycle pedal's path and speed without calculating how hard the rider must push. The geometry tells you where the pedal goes; the force calculation needs additional information.

The analogy highlights the boundary between describing motion and explaining its causes. Real mechanisms can flex, slip or have friction, so an ideal geometric model is only as accurate as its assumptions.

Worked example

Position, velocity and acceleration of one rotating link

Consider a rigid planar link of length 2 meters, hinged at the origin. Its angle θ is measured counterclockwise from the positive x axis. The tip position is x = 2 cos θ and y = 2 sin θ.

At θ = 30 degrees, the tip is at (√3, 1), approximately (1.732, 1) meters. This is a forward-kinematics calculation: the angle determines the endpoint location.

Suppose the angular velocity at that instant is 0.5 radians per second. Differentiating gives x-dot = −2 sin θ times θ-dot and y-dot = 2 cos θ times θ-dot. The tip velocity is therefore (−0.5, 0.866) meters per second, approximately. Its speed is the square root of 0.25 + 0.75, or 1 meter per second.

If that angular speed is constant, the tip acceleration is −0.25 times its position vector, approximately (−0.433, −0.25) meters per second squared. Its magnitude is 0.5 meters per second squared, directed inward. Constant speed has not meant zero acceleration: the direction of velocity is changing.

For an inverse question, a tip target at (0, 2) is achieved at 90 degrees, assuming the joint permits it. A target at (0, 3) is unreachable for this fixed-length link. None of these calculations determines the required motor torque; mass distribution, loads and other dynamic effects have not been specified. The derivative formulas use radians for angular rates even though the pose was described in degrees.

Illustrative case study

The fictional Willow Motion team creates a one-link teaching simulator. It correctly draws the endpoint but initially displays zero acceleration whenever the angular speed is constant.

The team checks the circular-motion calculation and changes the display to show inward acceleration. Reviewers also label angular position and angular velocity with their different units, preventing a degree-based input from being mistaken for a radian-based derivative.

When someone asks which motor the simulation requires, Willow explains that its geometric model has not supplied the needed load and inertia data. The fictional case demonstrates a useful kinematic model while keeping its force-related limitations visible.

Common misconceptions and mistakes

• Kinematics studies only positions. It also describes velocities and accelerations.
• Constant speed means zero acceleration. A changing direction produces acceleration even at constant speed.
• Inverse kinematics always has exactly one solution. Depending on the mechanism and target, it can have none, one or several.
• A geometrically reachable pose guarantees executable motion. Obstacles, constraints and actuator capabilities require further checks.

Business use

Kinematics helps teams test geometric feasibility, connect sensors to robot models and understand relationships between joint motion and task motion. It is also a practical foundation for simulation and animation workflows.

Before relying on results, confirm geometry, calibration, frames and units. A correct formula with an incorrect link length or joint zero can still predict the wrong location. Separate endpoint reachability from path planning and from force requirements when discussing project scope. That distinction helps a team ask for the right next calculation instead of treating one successful pose computation as a complete deployment assessment.

People also ask

Can kinematics include acceleration while ignoring force?

Yes. Acceleration describes motion; dynamics relates it to forces, torques and inertia.

What does a Jacobian do in velocity kinematics?

It maps joint rates to the selected end-effector velocity representation at a particular configuration.

Does a kinematic model need machine learning?

No. Many models follow directly from geometry. Learning can assist modeling or calibration, but it is not part of the definition.

Check your understanding

Why does the rotating link's tip accelerate even though its speed stays at 1 meter per second?

Answer: Its velocity direction changes continuously around the circle. Acceleration measures change in the velocity vector, including direction, so an inward acceleration remains.

Related terms

• Inverse kinematics
• Kinematic state
• Jacobian
• Pose
• Trajectory
• Forward kinematics
• angular velocity
• acceleration
• dynamics
• joint constraint

Learn next

Repeat the one-link calculation at 90 degrees with the same angular rate. Check that velocity is tangent to the circle and acceleration points inward. Then list the extra quantities required to estimate torque, keeping them separate from the geometric inputs.

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Sources

Lynch and Park introduce velocity kinematics and the Jacobian: https://modernrobotics.northwestern.edu/nu-gm-book-resource/velocity-kinematics-and-statics/

Their inverse-kinematics explanation distinguishes forward and inverse questions and possible solution counts: https://modernrobotics.northwestern.edu/nu-gm-book-resource/inverse-kinematics-of-open-chains/

Their constraints discussion explains restrictions on configurations and velocities: https://modernrobotics.northwestern.edu/nu-gm-book-resource/2-4-configuration-and-velocity-constraints/

The one-link calculations and Willow Motion case are original illustrations.

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