Unit 3: Robot Coordinate System - Subjective Questions

Answer:

A robot coordinate system is a reference frame used to describe the position and orientation of a robot link, joint, tool, or object in three-dimensional space. In NINO V2, coordinate systems help relate the humanoid's body, arms, gripper, and surrounding workspace.

The coordinate system is important because it:

• Specifies the location of the gripper relative to the robot body or world.
• Defines the orientation of the gripper for grasping and manipulation.
• Allows motion commands to be expressed mathematically.
• Provides a common reference for verifying robot movements.
• Helps convert measurements between the gripper coordinate system and the world coordinate system.

A pose is generally represented by a position vector and an orientation matrix, such as and .

Answer:

The world coordinate system is a fixed reference frame attached to the laboratory environment, robot base, or selected calibration point. It remains stationary while the robot moves.

The gripper coordinate system is a local reference frame attached to the gripper or end-effector. It moves and rotates with the gripper.

The main differences are:

• The world frame describes the gripper pose with respect to the environment.
• The gripper frame describes points and directions relative to the gripper itself.
• World-frame coordinates remain fixed during robot motion.
• Gripper-frame coordinates change as the gripper translates or rotates.
• A transformation matrix is required to convert a point from one frame to the other.

For example, an object located in front of the gripper may have constant gripper-frame coordinates, even though its world-frame coordinates change when NINO V2 moves.

Answer:

The position and orientation of the gripper can be represented using a homogeneous transformation matrix:

Here:

• denotes the world coordinate system.
• denotes the gripper coordinate system.
• is the rotation matrix describing gripper orientation relative to the world.
• is the position vector of the gripper origin in the world frame.
• The final row enables translation and rotation to be handled in one matrix operation.

A point expressed in gripper coordinates can be converted into world coordinates by multiplying it by this transformation matrix in homogeneous form.

Answer:

Let a point be represented in the gripper frame as . The homogeneous representation is:

The transformation from the gripper frame to the world frame is:

Therefore, the point in world coordinates is:

Expanding the multiplication gives:

The rotation matrix first rotates the point according to the gripper orientation, and the position vector then translates it to the gripper origin in the world frame.

Answer:

If the transformation from the gripper frame to the world frame is , then the reverse transformation is its inverse:

For a rigid-body transformation:

Thus, a world-frame point is converted using:

The translation is removed first, and the remaining vector is rotated using the transpose of the rotation matrix. The transpose is used because a rotation matrix is orthonormal.

Answer:

A suitable verification procedure is:

• Establish the world coordinate frame using the robot base, a calibration fixture, or marked reference points.
• Record the commanded gripper pose from the NINO V2 control interface.
• Identify the gripper origin and selected reference points on the gripper.
• Calculate the expected world coordinates using .
• Move the gripper to the commanded pose at low speed.
• Measure the actual position using a ruler, calibration board, motion-capture system, or known markers.
• Compare the calculated and measured positions along the , , and axes.
• Repeat the measurement for several poses.

The position error can be calculated as:

Small and repeatable errors indicate that the position transformation is correctly implemented.

Answer:

The gripper coordinate system normally contains three mutually perpendicular unit axes: , , and . These axes can be represented as the columns of the rotation matrix:

Orientation verification can be performed as follows:

• Mark or identify the positive axes on the gripper.
• Command NINO V2 to a known orientation.
• Observe the directions of the gripper axes relative to the world axes.
• Compare the observed axis directions with the columns of the calculated rotation matrix.
• Confirm that each axis has unit length and that the axes remain perpendicular.

The rotation matrix should satisfy:

These conditions confirm that the matrix represents a valid three-dimensional rotation.

Answer:

Position transformation determines where the gripper origin is located relative to another coordinate frame. It is represented by a translation vector:

Orientation transformation determines how the gripper axes are rotated relative to another coordinate frame. It is represented by a rotation matrix, Euler angles, or a quaternion.

The distinction is:

• Position describes location in space.
• Orientation describes angular alignment.
• Position uses units such as millimetres or metres.
• Orientation uses degrees, radians, rotation matrices, or quaternions.
• Translation changes the origin but not the direction of the axes.
• Rotation changes the direction of the axes but does not necessarily change the origin.

A homogeneous transformation combines both aspects so that the complete pose of the NINO V2 gripper can be described by one matrix.

Answer:

The three representations have different properties:

• Euler angles: Represent orientation using three successive rotations, such as roll, pitch, and yaw. They are easy to understand but depend on rotation order and may suffer from gimbal lock.
• Rotation matrices: Use a matrix to transform vectors between frames. They are direct and convenient for calculations, but contain nine values with constraints.
• Quaternions: Use four parameters to represent orientation. They avoid gimbal lock and are efficient for interpolation, but are less intuitive and must be normalized.

For laboratory verification, rotation matrices are especially useful because their columns directly show the gripper axes in the world frame. Euler angles may be convenient for displaying results, while quaternions are useful for smooth robot motion and software implementation.

Answer:

A coordinate-frame convention specifies the direction of each axis, the origin, and the positive sense of rotation. Consistent conventions are essential because the same numerical values can represent different physical poses under different conventions.

During NINO V2 verification, the following must be specified:

• The location of the world-frame origin.
• The positive directions of the world , , and axes.
• The location and orientation of the gripper-frame origin.
• Whether vectors are treated as column or row vectors.
• The order of elementary rotations.
• The units used for distance and angle.

For example, using a rotation order of produces a different orientation from . Any mismatch between the robot software and the laboratory calculations can produce apparently incorrect transformation results.

Answer:

The translation of the gripper origin is represented by:

Let the gripper orientation relative to the world be represented by:

Combining rotation and translation gives:

This matrix transforms a homogeneous point from the gripper frame into the world frame and completely describes the gripper pose.

Answer:

A known calibration point provides an independent reference for testing the transformation.

The procedure is:

• Attach or identify a point on the gripper whose coordinates in the gripper frame are known as .
• Obtain the gripper pose from the NINO V2 system.
• Calculate the predicted world position using:

• Measure the actual world position of the calibration point.
• Calculate the position error between the predicted and measured values.
• Repeat the test at different gripper orientations and locations.

If the error remains within the allowable tolerance, the transformation is verified. Large errors may indicate incorrect frame definitions, calibration errors, encoder offsets, mechanical backlash, or unit mismatches.

Answer:

The experimental verification can be organized into the following steps:

• Inspect the robot and ensure that the gripper, joints, and reference markers are correctly identified.
• Define the world and gripper coordinate systems according to the laboratory convention.
• Calibrate the robot or record any available zero offsets.
• Select several test poses that include translations and rotations about different axes.
• Record the commanded pose and the transformation matrix for each pose.
• Compute the expected location of marked gripper points in the world frame.
• Measure the actual point locations and gripper-axis directions.
• Compare predicted and measured positions and orientations.
• Calculate position error and orientation error.
• Repeat measurements to evaluate accuracy and repeatability.

A complete verification should test independent movements along all three axes and rotations about all three axes, while observing safety limits and avoiding singular or unreachable configurations.

Answer:

Accuracy is the closeness of the measured gripper pose to the commanded or theoretically calculated pose. A robot may be repeatable but inaccurate if it reaches nearly the same incorrect location every time.

Repeatability is the ability of NINO V2 to return to the same pose under the same conditions. It is evaluated by recording the spread of repeated measurements.

For position, accuracy can be expressed as:

Repeatability can be estimated from the variation or standard deviation of repeated positions. Accuracy is affected by calibration and model errors, while repeatability is affected by backlash, mechanical flexibility, sensor noise, and controller consistency.

Answer:

A transformation matrix contains both rotation and translation information. Errors in either part affect the calculated pose.

• Translation errors shift the calculated gripper origin directly along one or more world axes.
• Rotation errors change the directions of the gripper axes.
• A small orientation error can produce a large position error for a point located far from the gripper origin.
• Incorrect frame assignment can cause systematic errors in every measurement.
• Non-orthogonal rotation matrices may introduce scaling or distortion, which is physically invalid for a rigid body.

For a point at a distance from the gripper origin, the effect of a small angular error is approximately proportional to the distance from the origin:

Therefore, both origin position and axis orientation must be checked during NINO V2 laboratory verification.

Answer:

The columns of the rotation matrix describe the unit axes of the gripper frame expressed in world coordinates:

Thus:

• The first column gives the direction of the gripper -axis in the world frame.
• The second column gives the direction of the gripper -axis in the world frame.
• The third column gives the direction of the gripper -axis in the world frame.

This interpretation is useful for verifying orientation. If the gripper is commanded to point its -axis upward, the third column should approximately align with the positive world -axis. The columns must also be mutually perpendicular and have unit magnitude.

Answer:

The gripper reference point defines the origin of the gripper coordinate system. Changing this point changes the translation component of the transformation, even if the physical gripper orientation remains unchanged.

If the new reference point has coordinates relative to the original gripper origin, its world position is:

The orientation matrix is unchanged if the new frame has the same axis directions. However, the position must be recalculated because the offset vector is rotated into the world frame.

This is important in NINO V2 because the controller's tool center point may not coincide with the physical centre of the gripper. Using an incorrect reference point produces a systematic position error.

Answer:

A transformation between successive frames can be built by multiplying elementary homogeneous transformations. A translation along the axes is represented by:

For a rotation about the -axis by angle :

If the translation is followed by this rotation, the combined transformation is:

The order of multiplication is significant because three-dimensional rotations and translations generally do not commute.

Answer:

Homogeneous transformations represent physical operations such as translation and rotation. In three-dimensional space, performing two operations in different orders generally produces different final poses.

For example:

• Translating a gripper and then rotating it can move the origin along a different path from rotating first and then translating.
• Rotating about a world axis is different from rotating about a gripper-fixed axis.
• The product is generally not equal to .

Therefore, the analyst must identify whether each operation is defined relative to a fixed world frame or a moving gripper frame. Incorrect multiplication order can cause both position and orientation verification to fail even when the individual matrices are correct.

Answer:

Position error is calculated by subtracting the reference position from the measured position:

The magnitude of the position error is:

Orientation error can be obtained from the relative rotation matrix:

The equivalent angular error is:

The calculated errors should be compared with the permitted laboratory tolerance. Repeated tests help determine whether the errors are systematic or random.