Unit 4: Robot Mechanical Design Parameters - Subjective Questions

Robot kinematics is the study of the motion of a robot without considering the forces and torques that cause the motion. It establishes the relationship between the joint variables and the position and orientation of the robot end-effector.

The main aspects of robot kinematics are:

• Forward kinematics: Determines the end-effector pose from known joint variables.
• Inverse kinematics: Determines the joint variables required to achieve a specified end-effector pose.
• Velocity kinematics: Relates joint velocities to end-effector linear and angular velocities.
• Trajectory analysis: Describes how the robot moves between two or more configurations.

Kinematics is important because it is used for robot programming, path planning, workspace analysis, collision avoidance, and position control. For a manipulator with joint vector , the end-effector pose can generally be written as , where is the forward kinematic function.

A coordinate frame is a reference system consisting of an origin and mutually perpendicular axes used to specify the position and orientation of a robot link or object. Common frames include the world frame, base frame, joint frames, link frames, and end-effector frame.

Coordinate transformations are required because different robot components are described relative to different reference frames. A transformation allows the position and orientation of an object expressed in one frame to be represented in another frame.

A homogeneous transformation matrix is commonly written as:

where represents the rotation of frame relative to frame , and represents the position of the origin of frame expressed in frame . Transformations are essential for computing the end-effector pose, combining link motions, and controlling industrial robots.

A homogeneous transformation matrix combines rotation and translation into a single matrix. It is expressed as:

Here, is a rotation matrix and is a position vector. The final row allows translation and rotation to be combined through matrix multiplication.

For a point represented in frame , its coordinates in frame are obtained using:

The transformation between the base and end-effector frames is obtained by multiplying the transformations of individual links:

Thus, homogeneous transformations provide a systematic method for calculating the complete pose of a robot end-effector.

Degrees of freedom (DOF) are the number of independent variables required to completely define the configuration of a robot. A rigid body moving freely in three-dimensional space has six DOF:

• Three translational DOF along the , , and axes.
• Three rotational DOF about the , , and axes.

The DOF of a robot is usually determined by its joints. A revolute joint generally contributes one rotational DOF, while a prismatic joint contributes one translational DOF. Therefore, a serial robot with independent single-DOF joints normally has DOF.

A robot must have enough DOF to perform its intended task. A six-axis manipulator can independently control the complete position and orientation of an object in three-dimensional space. Robots with fewer than six DOF may be unable to achieve arbitrary poses, while robots with more than six DOF are redundant and can use the extra DOF for obstacle avoidance or improved posture.

The pose of a robot end-effector consists of its position and orientation relative to a reference frame. Position is represented by a vector:

Orientation describes how the end-effector axes are rotated relative to the reference axes.

Common orientation representations include:

• Rotation matrices: A orthonormal matrix with determinant equal to one.
• Euler angles: Three successive rotations about specified axes, such as roll, pitch, and yaw.
• Axis-angle representation: A rotation through angle about a unit vector .
• Quaternions: A four-parameter representation that avoids some singularities associated with Euler angles.

The complete pose can be represented using a homogeneous transformation matrix. Accurate pose representation is necessary for robot positioning, welding, assembly, machining, and grasping operations.

Forward kinematics determines the position and orientation of the end-effector when the joint variables and robot geometry are known.

The general procedure is:

• Assign coordinate frames to the robot links using a consistent convention, such as the Denavit-Hartenberg convention.
• Define the joint variables and geometric parameters.
• Construct the homogeneous transformation matrix for each consecutive pair of frames.
• Multiply the individual matrices in sequence.
• Extract the position vector and rotation matrix from the final transformation matrix.

For a serial manipulator with joints:

The resulting matrix is:

where gives the end-effector position and gives its orientation.

Forward kinematics maps the joint configuration of a robot to the pose of its end-effector. If the joint vector is , the pose is represented as:

The input consists of joint angles for revolute joints and joint displacements for prismatic joints. The output is the position and orientation of the tool center point.

Important applications include:

• Determining the location of a welding torch or cutting tool.
• Simulating robot motion before executing it on the physical machine.
• Checking whether the robot can reach a workpiece.
• Monitoring the position of the end-effector using joint sensor data.
• Generating graphical representations of robot configurations.
• Supporting collision detection and workspace analysis.

Forward kinematics normally provides a unique end-effector pose for a given joint configuration, although different joint configurations may sometimes produce the same pose in robots with special geometries.

Inverse kinematics determines the joint variables required to place the robot end-effector at a specified position and orientation. It is the reverse mapping of forward kinematics.

The relationship can be written as:

The comparison is as follows:

• In forward kinematics, joint variables are known and the end-effector pose is calculated.
• In inverse kinematics, the desired end-effector pose is known and joint variables are calculated.
• Forward kinematics is generally direct and has one result for a particular configuration.
• Inverse kinematics may have multiple solutions, no solution, or infinitely many solutions.
• Inverse kinematics is usually more mathematically complex.

Inverse kinematics is required for robot programming because users generally specify task locations in Cartesian coordinates, while robot controllers must command individual joints.

Inverse kinematics may produce different types of solutions depending on the robot geometry and desired pose.

• Multiple solutions: A robot may reach the same pose with different configurations, such as elbow-up and elbow-down or left-arm and right-arm postures.
• No solution: The desired pose may lie outside the robot workspace or may require an orientation that the mechanism cannot achieve.
• Unique solution: Some poses and robot geometries produce only one valid configuration.
• Infinite solutions: Redundant robots may have extra joints, allowing many joint configurations to produce the same end-effector pose.

A singular configuration occurs when the robot loses one or more instantaneous motion directions. At such a configuration, small Cartesian motions may require very large joint velocities or may be impossible. Singularities are associated with a rank-deficient Jacobian and must be considered during inverse kinematics and trajectory planning.

Velocity kinematics describes the relationship between the rates of change of joint variables and the linear and angular velocity of the end-effector.

If is the joint vector and represents the end-effector pose, differentiating the forward kinematic relationship gives:

where:

• is the end-effector velocity vector.
• is the joint velocity vector.
• is the robot Jacobian matrix.

For a spatial manipulator, the velocity vector is commonly written as:

where is linear velocity and is angular velocity. The Jacobian is obtained from the partial derivatives of the forward kinematic equations:

The Jacobian is used in velocity control, singularity analysis, force transformation, and motion planning.

The Jacobian matrix is a matrix that maps joint velocities to end-effector linear and angular velocities:

Its roles include:

• Computing the instantaneous velocity of the end-effector.
• Converting desired Cartesian velocities into joint velocities.
• Relating forces and torques between joint space and Cartesian space.
• Detecting singular robot configurations.
• Supporting resolved-rate motion control.

A robot is in a singular configuration when the Jacobian loses rank. For a square Jacobian, this can occur when:

At a singularity, some end-effector motions cannot be generated independently, and inverse velocity calculations may require extremely large joint velocities. Therefore, trajectory planners and controllers try to avoid singular configurations or use special numerical methods near them.

A robot path is the geometric route followed by the end-effector or robot joints, without specifying how fast the route is traversed. A trajectory includes both the path and the time history of motion.

For example, a Cartesian path may be defined by a sequence of points , while a trajectory specifies the corresponding functions and .

Basic requirements of trajectory planning include:

• Passing through the required starting, intermediate, and final points.
• Maintaining smooth position, velocity, and acceleration profiles.
• Respecting joint position, velocity, and acceleration limits.
• Avoiding collisions and robot singularities.
• Minimizing excessive energy use, vibration, or actuator effort.
• Synchronizing the motion of multiple joints.

Common interpolation methods include polynomial, linear, and spline interpolation in joint space or Cartesian space.

Robot path planning determines a collision-free and feasible route from an initial configuration to a desired configuration or task location.

In joint-space planning, the joint variables are directly interpolated. For example:

Advantages include simple computation, easy enforcement of joint limits, and smooth actuator commands. However, the end-effector path may not be a straight line in Cartesian space.

In Cartesian-space planning, the desired end-effector position and orientation are interpolated directly. Inverse kinematics is then used to calculate joint values. This method is useful for welding, painting, and machining where the tool must follow a specified geometric path.

A practical planner must consider workspace obstacles, robot joint limits, velocity and acceleration constraints, singularities, and the required accuracy of the task.

Robot dynamics studies the relationship between the motion of a robot and the forces and torques that produce that motion. It considers mass, inertia, gravity, friction, external loads, joint accelerations, and actuator efforts.

The difference between kinematics and dynamics is:

• Kinematics describes position, orientation, velocity, and acceleration without considering forces.
• Dynamics determines the forces and torques required to produce a specified motion or predicts the motion produced by applied forces.

A general robot dynamic model can be represented as:

where is the vector of joint torques, is the inertia matrix, represents Coriolis and centrifugal effects, represents gravity effects, represents friction, and represents external load effects.

The Newton-Euler formulation derives robot dynamics by applying Newton's second law to the translational motion of each link and Euler's rotational equation to its rotational motion.

For a link, the translational equation is:

and the rotational equation is:

where is force, is mass, is linear acceleration, is moment, is the inertia tensor, is angular acceleration, and is angular velocity.

The recursive Newton-Euler algorithm generally has two stages:

• Forward recursion: Calculates link velocities and accelerations from the base toward the end-effector.
• Backward recursion: Calculates forces, moments, and joint torques from the end-effector toward the base.

Its main advantages are computational efficiency, suitability for real-time control, and direct physical interpretation. It is widely used for inverse dynamic calculations in industrial robot controllers.

The Lagrangian formulation derives robot equations of motion using energy rather than directly summing forces and moments. The Lagrangian is defined as:

where is the total kinetic energy and is the total potential energy of the robot.

The equation for joint is:

where is a generalized coordinate, is its velocity, and is the corresponding generalized force or torque.

After applying this equation to all joints, the robot dynamics commonly takes the form:

The Lagrangian method is systematic and convenient for deriving symbolic equations, especially for robots with complex link structures. However, the resulting equations may become computationally intensive for robots with many joints.

Both Newton-Euler and Lagrangian formulations describe the relationship between robot motion and actuator forces or torques, but they use different approaches.

The Newton-Euler method is often preferred for high-speed computation, while the Lagrangian method is useful for obtaining theoretical equations and studying system properties.

Forces and torques in a robotic manipulator arise from the motion of its links, gravity, friction, external loads, and contact with objects.

The actuator torque can be represented by:

The major components are:

• Inertial torque: Required to produce joint acceleration.
• Coriolis and centrifugal torques: Produced by the interaction of joint velocities and link motion.
• Gravity torque: Required to support the weight of links and payloads.
• Friction torque: Caused by losses in bearings, gears, and transmissions.
• External torque: Caused by contact forces, payloads, or interaction with the environment.

Accurate estimation of these forces and torques is important for motor selection, controller design, safety analysis, force control, and preventing structural overload.

Static balance means that a robot remains in equilibrium when it is stationary. The sum of all external forces and moments must be zero:

Static balance is affected by the robot's center of gravity, support polygon, payload, and joint configuration. A stationary robot may become unstable if its center of gravity moves outside the support region.

Dynamic balance considers motion and acceleration. In addition to gravitational forces, inertial forces and moments are present. Dynamic balance requires controlling these effects so that vibrations, structural loads, and base reactions remain acceptable.

Balance affects mechanical design in several ways:

• The base must be sufficiently stable and rigid.
• Heavy components may be placed near the base.
• Counterweights or balancing mechanisms may be used.
• Link masses and moments of inertia should be minimized.
• Motion profiles should limit sudden acceleration and deceleration.

Good balance improves accuracy, safety, energy efficiency, and robot service life.

For a manipulator in static equilibrium, the virtual work performed at the joints must equal the virtual work performed by an external end-effector wrench. Let be a virtual joint displacement and the corresponding end-effector displacement.

The velocity relationship is:

If is the end-effector wrench and is the joint torque vector, equality of virtual work gives:

Substituting the displacement relationship:

Since this must hold for any :

This relationship allows a desired Cartesian force to be converted into joint torques. It is used in force control, compliant assembly, polishing, grinding, human-robot interaction, and estimating the effect of payload forces on actuators.