# Can a 6 DOF robot have multiple inverse kinematics solution with a constrained rotation/orientation

I trying to understand how manipulators robot works (6 DOF)

I would like to know if the inverse kinematics (IK) solution give only one or multiple solution if we include the rotation (3x3 matrix) of the TCP ? In others words, can we have the same TCP position and orientation for multiple configuraiton angles of the robot ?

I know that forward kinematics (FK) give only 1 solution (which seems logic) but IK is way more difficult to represent.

• There are still a probability that position and orientation you desired would have several possible solution or maybe no solution at all Mar 25, 2021 at 14:42

can we have the same TCP position and orientation for multiple configuraiton angles of the robot

Sure!

The number of DOF of a manipulator tells us if it's possible or not to reach a pose (position + orientation) in the operational space at hand (usually 6-dimensional).

## Examples

• If we work in a 6D operational space, then we are required to have at least a 6-DOF manipulator to reach a full pose (when this pose is within the allowed range).
• If we deal with a planar manipulator and we are interested in attaining only a target position (i.e. no orientation), then we need a 2-DOF manipulator at least.
• Same as above but now we include the orientation of the gripper; it follows that we need at least a 3-DOF manipulator.

Notice how I've highlighted the term "at least".

On the other hand, nothing is said about how many joint configurations can resolve the single given target pose.

In general, we have many joint configurations! A clear example of that is provided by a 6-DOF manipulator that can easily move all its joints while keeping the position as well as the orientation of the tip fixed. The latter can be achieved by resorting to the so-called null-space projection method.

Absolutely it can. Up to 16 solutions may exist for a generic 6 axis robot. If the last three axes intersect in a point, the maximum number of solutions is reduced to 8.

Some of these solutions may be outside the joint limits, where the particular robot cannot specifically rotate. This may restrict the number of possible solutions, at time limiting it to one or also may happen that to none. This depends on the robot, pose and constraints. See this article for more.

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