Unit 4: Forward Kinematics - Subjective Questions

Forward kinematics is the process of determining the position and orientation of a robot's end effector from the known joint variables and geometric parameters of the robot.

For a two-axis planar articulated robot:

• The joint variables are usually the joint angles and .
• The link lengths are represented by and .
• The robot operates in a two-dimensional plane.
• The output is the end-effector position and its orientation relative to the base.

Forward kinematics is useful for:

• Predicting the location of the gripper for given motor angles.
• Controlling and programming the robot.
• Verifying the accuracy of the robot model.
• Simulating robot motion before applying commands to the physical Orangewood Robotic Arm.
• Converting joint-space commands into Cartesian coordinates.

A two-axis planar articulated robot consists of two rotary joints connected by two rigid links. The first joint connects the base to the first link, and the second joint connects the first link to the second link or end effector.

The main components are:

• Base: Provides mechanical support and defines the reference coordinate frame.
• Joint 1: Rotates the first link by angle .
• Link 1: Has length and rotates about the base joint.
• Joint 2: Rotates the second link relative to the first link by angle .
• Link 2: Has length and carries the end effector.
• End effector: The point whose position is calculated using forward kinematics.

Both joints rotate about axes perpendicular to the plane of motion. The absolute orientation of the second link is . By changing the two joint angles, the end effector can reach different points within the robot's workspace.

Consider a two-link planar robot with link lengths and , and joint angles and . The first link makes an angle with the positive -axis. The second link is oriented at the absolute angle .

The coordinates contributed by the first link are:

The coordinates contributed by the second link are:

Adding the contributions gives the end-effector coordinates:

The end-effector orientation is:

These equations are the analytical forward kinematic model of the two-axis planar articulated robot.

A relative joint angle describes the rotation of one link with respect to the previous link. An absolute link orientation describes the angle of a link with respect to the fixed base coordinate frame.

For a two-axis planar robot:

• is both the first joint angle and the absolute orientation of the first link.
• is the rotation of the second link relative to the first link.
• The absolute orientation of the second link is the sum of the two joint angles:

Therefore, the second link does not generally point at angle with respect to the base. It points at angle .

This distinction is important because the second link's contribution to the end-effector position must be calculated using and .

The analytical forward kinematic equations are:

Substitute the given values:

Therefore,

Similarly,

Hence, the end-effector position is approximately:

The end-effector orientation is .

The Denavit-Hartenberg convention is a systematic method for assigning coordinate frames to robot links and describing the relative transformation between consecutive frames using four parameters.

The four standard DH parameters are:

• or link length: The distance between the and axes measured along the axis.
• or link twist: The angle between the and axes measured about the axis.
• or link offset: The distance along the axis from the origin of frame to the intersection with the axis.
• or joint angle: The rotation about the axis from the axis to the axis.

For a revolute joint, is variable and the other parameters are generally constant. The DH convention simplifies a robot's geometry into a sequence of homogeneous transformation matrices.

For a two-link planar robot, the joint axes are parallel and perpendicular to the plane of motion. A commonly used standard DH table is:

The interpretation is:

• The first link has length and joint variable .
• The second link has length and joint variable .
• The twist angles are zero because all joint axes are parallel.
• The link offsets are zero because the joints are located in the same plane without displacement along the joint axes.

Depending on the exact frame assignment and DH convention used by the laboratory manual, signs or frame labels may differ. However, a correct table must produce the same physical end-effector coordinates.

The standard DH transformation from frame to frame is:

The matrix contains:

• A rotation component that describes the orientation of frame relative to frame .
• A translation component that describes the position of the new frame origin.
• The parameters , , , and , which describe the link geometry and joint configuration.
• The last row , which allows rotation and translation to be represented together using homogeneous coordinates.

For the first link, use the DH parameters:

Substituting these values into the standard DH matrix gives:

This transformation represents:

• A rotation of about the base -axis.
• A translation of along the rotated -axis.
• No twist between the two frames because .
• No offset along the -axis because .

The origin of frame 1 is therefore located at:

For the two-axis planar robot, the individual transformations are:

The total transformation is:

After multiplication:

The fourth column gives the end-effector position, while the upper-left block gives its orientation.

Both methods calculate the end-effector pose, but they differ in procedure and generality.

For the Orangewood two-axis planar arm, the analytical method is convenient for quickly deriving:

The DH method is valuable because the same modeling approach can later be extended to robots with more joints or three-dimensional motion.

Coordinate-frame assignment is performed systematically as follows:

• Assign the base frame to the robot base.
• Place each joint axis along the corresponding -axis.
• For a planar robot, the joint axes are parallel and perpendicular to the plane of motion.
• Place the -axis along the common normal between consecutive -axes. For a planar arm, this generally aligns with the link.
• Choose the -axis to complete a right-handed coordinate system.
• Locate each frame origin at the intersection of the joint axis and the selected -axis, or at the common normal location.
• Assign the final frame to the end effector.

For the two-axis planar robot, the twist angles are typically zero because the joint axes are parallel. The link lengths are represented by the parameters, and the revolute joint positions are represented by .

The orientation of the first link relative to the base is . The second joint rotates the second link by relative to the first link. Therefore, the absolute orientation of the end effector is:

This means that the end-effector frame is rotated by with respect to the base frame. The corresponding planar rotation matrix is:

Physically, specifies the direction in which the gripper or tool is pointing. It is different from the position , which specifies the location of the end effector.

A homogeneous transformation matrix combines the rotation and translation between two coordinate frames into one mathematical representation.

A general transformation has the form:

where:

• is a rotation matrix.
• is a position vector.
• The final row enables rotation and translation to be multiplied as a single matrix operation.

In forward kinematics, transformations for successive links are multiplied in order:

The final matrix gives the position and orientation of the end-effector frame with respect to the base frame. This approach avoids repeatedly treating rotations and translations as separate calculations.

The final orientation is:

The position is obtained from the fourth column of :

Substituting the values:

Thus, the final pose is approximately:

The corresponding homogeneous transformation is:

The end-effector position is:

Changing rotates the entire arm about the base. Therefore:

• Both links change their absolute orientations.
• The end effector generally moves along a large circular path about the base.
• The overall orientation of the second link changes by the same amount.

Changing rotates only the second link relative to the first link. Therefore:

• The first link remains fixed.
• The end effector moves along a circular path centered approximately at the second joint.
• The second-link orientation changes directly.

The joint angles influence both position and orientation, while the link lengths determine the size and reach of the workspace.

The forward kinematic equations are:

The position Jacobian relates joint velocities to Cartesian velocities:

Calculate the partial derivatives:

Hence,

This matrix describes how small changes in joint angles affect the end-effector position.

A singular configuration is a robot posture in which the robot loses one or more instantaneous directions of motion. At such a configuration, independent joint movements cannot produce independent Cartesian movements.

For the two-link planar robot, the determinant of the position Jacobian is:

Therefore, singularities occur when:

which gives:

These configurations correspond to the links being fully stretched or folded along the same line. The robot then loses motion authority in one Cartesian direction.

Although forward kinematics still produces a valid position at a singularity, the robot's local motion behavior becomes problematic. In practice, singularities can cause large joint velocities, poor control accuracy, or unstable inverse-kinematic calculations.

The reachable workspace is the set of all points that the end effector can reach as the joint angles vary within their allowable limits.

For link lengths and , the maximum radial reach is:

If the second link folds back against the first, the minimum radial distance from the base is:

With unrestricted revolute joints, the workspace is an annular region bounded by these radii:

In a real Orangewood Robotic Arm:

• Joint limits may reduce the workspace to a sector or partial annular region.
• Mechanical interference may restrict some positions.
• The gripper dimensions may reduce the usable workspace.
• Calibration errors may cause the actual workspace to differ from the theoretical model.

The forward kinematic equations map every allowed pair to a point in this workspace.

Correct angle handling is essential because trigonometric functions and physical joint conventions must agree.

Important considerations include:

• Most programming languages expect angles in radians, even when the user enters values in degrees.
• Conversion from degrees to radians is:

• A positive angle must be defined consistently, usually counterclockwise when viewed along the positive -axis.
• A negative angle represents rotation in the opposite direction.
• The second joint angle is usually measured relative to the first link, not directly from the base.
• Encoder zero positions may not match the mathematical zero positions.
• Joint offsets may be required to relate encoder readings to model angles:

Incorrect units or sign conventions can produce a mathematically consistent but physically incorrect end-effector position.