2.2. Redundant Manipulators and Closed Kinematic Chains

A manipulator is redundant when it has more degrees of freedom (DOF) than required to perform a task.

For example:

• A 7-DOF robot arm operating in 3D space (which needs only 6 DOF for position + orientation) is redundant.

Advantages:

• Greater flexibility in motion.
• Can avoid obstacles or optimize posture.
• Allows joint limit avoidance and energy efficiency.

Mathematical Implication:

• The IK problem becomes underdetermined (more unknowns than equations).
• Solutions exist in an infinite space.

Redundant manipulators refer to robotic systems that possess more degrees of freedom (DOF) than necessary for completing a specified task. For instance, in a 3D operational space, a robot arm may have 7 DOF while only needing 6 to adequately control its position and orientation. This excess capacity allows the robot to perform tasks in a more versatile and efficient manner.

Advantages of redundant manipulators include enhanced flexibility in movement, the ability to avoid obstacles, optimization of posture for more efficient energy consumption, and the capability to navigate around joint limits. By possessing more DOF, such robots can find various configurations to achieve the same end goal, making them adaptable.

From a mathematical perspective, the inverse kinematics (IK) problem becomes underdetermined with redundancy, suggesting that there are more unknowns (the various joint movements) than equations (the constraints needed to determine those movements). Thus, a multitude of possible solutions exist, rendering it easier for the robot to complete tasks in various configurations.

Think of a person walking through a crowded room. If they have fewer limbs, like one arm or one leg, they may struggle to navigate around obstacles effectively. However, if they have more than enough limbs — say, like an octopus — they can maneuver flexibly, using different arms to push aside or avoid obstacles. Similarly, a redundant manipulator has more joint movements available, allowing it to ‘navigate’ its task space more freely, optimizing its path and posture.

A closed kinematic chain is a structure where two or more links form a loop, creating multiple paths between two points.

Example: A parallel manipulator like the Stewart platform.

Properties:

• Higher mechanical stiffness.
• Better load-bearing capacity.
• Complex forward kinematics, but often simpler inverse kinematics.

Challenges:

• Requires constraint equations to maintain loop closure.
• Limited workspace compared to open chains.

Closed kinematic chains refer to a configuration where two or more links create a loop within the system. In this scenario, multiple paths link two points in the robotic structure, exemplified by a parallel manipulator like the Stewart platform, which features six actuated legs engaging a platform.

The main advantages of closed kinematic chains include increased mechanical stiffness, which ensures that the robot maintains its shape under load, and improved load-bearing capacity, allowing for greater weight to be handled effectively. Additionally, while the forward kinematics calculations can be complex due to the need to consider multiple links forming a loop, the inverse kinematics can often be simpler due to the loop providing more constraints.

However, there are challenges that come with closed kinematic chains. Maintaining loop closure requires specific constraint equations, and these constraints can limit the operational workspace. As a result, robots with closed kinematic chains may not be able to move as freely as their open-chain counterparts.

Consider a bicycle with a closed-loop chain system that connects the pedals to the rear wheel. The closed chain ensures that when you pedal, the motion is directly transferred to the wheel, creating a stiff and robust connection that allows for efficient movement. Just as a bicycle can transfer energy effectively through its closed-loop mechanism, closed kinematic chains in robotics allow for stable and efficient force transfer through interconnected links, although they might not be able to navigate as broadly through varying terrain as a freely pedaling robot on an open chain.

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