This revision fixes the planar-pair constraint set instead of letting a slider redefine the joint. The moving plate stays coincident with its reference plane while entered x, y and yaw define an exact planar pose.
Planar-Pair Pose Explorer
Enter a target x translation, y translation and yaw angle. The worksheet reports the target translation magnitude and rotation coefficients while the animation preserves z=0, roll=0 and pitch=0.
Equation Used
- Both ideal contact surfaces remain coincident in one plane.
- Relative z translation, roll and pitch are constrained to zero.
- Relative x translation, y translation and yaw about the plane normal remain free.
- The animation interpolates from the reference pose to the entered target pose.
- Both bodies are rigid and no deformation or surface error is modeled.
- Preload, gravity, friction, lubrication, clearance, flatness, stiffness, accuracy, wear, load capacity and safety are excluded.
Fixes the joint at its actual three-coordinate mobility and removes unsupported precision, preload, force, maintenance and failure guidance.
Three relative coordinates remain free
Modern Robotics shows that a spatial rigid body has six degrees of freedom and that three independent constraints reduce it to planar motion. The remaining coordinates are two translations in the plane and one angle about the normal.
The animation enforces exactly that set: x, y and yaw vary, while z, roll and pitch remain zero.
The ideal pair is a kinematic model
A planar pair describes allowable relative motion between ideal bodies. Actual implementations may use sliding contact, air films, magnetic or vacuum retention, or other structures, each with application-specific force, stiffness, surface and control requirements not determined by the pair classification.
Represent the relative pose in SE(2)
The translation vector needs two independent coordinates and the in-plane orientation needs one angle. Its magnitude is sqrt(x²+y²); cos(theta) and sin(theta) form the planar rotation matrix.
A declared planar pose
For x=60 mm, y=40 mm and yaw=30 degrees, target translation magnitude is 72.1 mm, yaw is 0.524 rad, cos(theta)=0.8660 and sin(theta)=0.5000. The pair still has three free relative coordinates.
Mobility does not establish implementation performance
The three-coordinate model says which ideal motions are allowed, not what forces retain contact or how accurately a real stage moves. Surface form, support, lubrication, contamination, thermal behavior, preload and controls require implementation-specific engineering evidence.
Unsupported micron flatness and film values, preload multipliers, pressure bands, brand claims, costs, load capacities, service lives, failure diagnoses, repair instructions and fabricated PCB examples are removed.
Planar-pair questions
Why are there three free coordinates?
Two locate the body in the plane and one sets its orientation about the normal.
Can the moving body lift or tilt?
Not in the ideal planar-pair model.
Does this calculate preload or friction?
No.
Does three DOF guarantee accuracy?
No.
Planar-pair references
- Modern Robotics, 2.1 Degrees of Freedom of a Rigid Body: six spatial freedoms, three independent planar freedoms and constraint counting.
- Carnegie Mellon, Basic Kinematics of Constrained Rigid Bodies: a plane pair removes three degrees of freedom in a spatial mechanism.
- Ansys Motion Theory, Constraints and Joints: plane-joint constraint and in-plane force/normal-torque freedom context.
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
