Calculations For Finding Cor Points

Calculate the Center of Rotation (CoR) for a rigid body based on the coordinates of two markers at two different time points or positions. Our Center of Rotation (CoR) calculation tool is easy to use.

CoR Calculator

Enter the coordinates of two markers (A and B) on a rigid body at two instances (Time 1 and Time 2).

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What is a Center of Rotation (CoR) calculation?

A Center of Rotation (CoR) calculation is a method used to determine the point around which a rigid body or segment appears to rotate during a given movement. In planar (2D) motion, if a body moves from one position to another, there’s a unique point (the CoR) that remains stationary relative to the body if the body were to rotate from the first position to the second. The Center of Rotation (CoR) calculation is crucial in fields like biomechanics, robotics, and mechanical engineering.

In biomechanics, the Center of Rotation (CoR) calculation helps analyze joint movement, such as the knee or shoulder, by identifying the effective pivot point. In engineering, it’s used to understand the kinematics of linkages and mechanisms. The instantaneous center of rotation (ICR) is a related concept for continuous motion.

Who should use it? Biomechanists, kinesiologists, engineers, animators, and anyone studying the motion of rigid bodies or segments connected by joints will find the Center of Rotation (CoR) calculation useful.

Common misconceptions include thinking the CoR is always within the physical joint or that it remains fixed throughout a large range of motion; in reality, for many biological joints, the CoR migrates as the joint moves, and we often calculate an average or instantaneous Center of Rotation (CoR) calculation for a specific movement phase.

Center of Rotation (CoR) calculation Formula and Mathematical Explanation

For a rigid body undergoing planar motion, identified by two markers (A and B) moving from position 1 (A1, B1) to position 2 (A2, B2), the Center of Rotation (CoR) is found at the intersection of the perpendicular bisectors of the displacement vectors A1A2 and B1B2.

Step-by-step derivation:

1. Midpoints: Find the midpoints of the segments A1A2 and B1B2.
   • M_A = ((Ax1 + Ax2)/2, (Ay1 + Ay2)/2)
   • M_B = ((Bx1 + Bx2)/2, (By1 + By2)/2)
2. Slopes of Segments: Calculate the slopes of A1A2 and B1B2.
   • m_A = (Ay2 – Ay1) / (Ax2 – Ax1)
   • m_B = (By2 – By1) / (Bx2 – Bx1)
   • Handle vertical lines (denominator = 0) separately.
3. Slopes of Perpendicular Bisectors: The slope of a perpendicular line is the negative reciprocal.
   • m_perpA = -1 / m_A (If m_A=0, m_perpA is undefined – vertical line; If m_A undefined, m_perpA=0)
   • m_perpB = -1 / m_B (If m_B=0, m_perpB is undefined; If m_B undefined, m_perpB=0)
4. Equations of Perpendicular Bisectors: Using point-slope form y – y1 = m(x – x1).
   • y – M_Ay = m_perpA * (x – M_Ax)
   • y – M_By = m_perpB * (x – M_Bx)
5. Intersection Point (CoR): Solve the system of two linear equations for x (CoR_x) and y (CoR_y). Special handling is needed for horizontal or vertical bisectors.

Variables Table

Variable	Meaning	Unit	Typical Range
Ax1, Ay1	Coordinates of Marker A at Time 1	Length (e.g., mm, cm, m)	Varies based on system
Bx1, By1	Coordinates of Marker B at Time 1	Length	Varies
Ax2, Ay2	Coordinates of Marker A at Time 2	Length	Varies
Bx2, By2	Coordinates of Marker B at Time 2	Length	Varies
CoRx, CoRy	Coordinates of the Center of Rotation	Length	Varies

The Center of Rotation (CoR) calculation provides the coordinates of this rotation point.

Practical Examples (Real-World Use Cases)

Example 1: Knee Joint Motion

In a biomechanics lab, markers are placed on the thigh and shank. At 30 degrees of knee flexion, marker A (on thigh) is at (100, 200) and B (on thigh) is at (100, 150). At 60 degrees, A moves to (110, 195) and B to (105, 145) relative to a fixed shank. Inputting these into the Center of Rotation (CoR) calculation can estimate the knee joint’s CoR for this range of motion.

• Ax1=100, Ay1=200, Bx1=100, By1=150
• Ax2=110, Ay2=195, Bx2=105, By2=145
• The calculator would output the CoR coordinates, giving insight into the knee’s pivot axis. This is vital for understanding joint kinematics.

Example 2: Four-Bar Linkage

An engineer is analyzing a four-bar linkage. Two points on one link are tracked: A1(0,0), B1(5,0) and A2(-1,1), B2(3.5, 3.5). The Center of Rotation (CoR) calculation for this link’s movement from position 1 to 2 helps determine the instantaneous center of rotation, which is important for velocity and acceleration analysis in rigid body motion.

• Ax1=0, Ay1=0, Bx1=5, By1=0
• Ax2=-1, Ay2=1, Bx2=3.5, By2=3.5
• The resulting CoR provides a point for further kinematic analysis.

Using a Center of Rotation (CoR) calculation is standard in these analyses.

How to Use This Center of Rotation (CoR) calculation Calculator

1. Enter Marker Coordinates at Time 1: Input the X and Y coordinates for Marker A (Ax1, Ay1) and Marker B (Bx1, By1) at the initial position or time.
2. Enter Marker Coordinates at Time 2: Input the X and Y coordinates for Marker A (Ax2, Ay2) and Marker B (Bx2, By2) at the final position or time. Ensure the units are consistent for all coordinates.
3. View Results: The calculator automatically updates and displays the CoR coordinates (CoR X, CoR Y), midpoints, and slopes as you enter valid numbers.
4. Interpret Results: The “Primary Result” shows the calculated CoR coordinates. The “Intermediate Results” provide values used in the calculation, which can be useful for understanding the geometry.
5. Use the Chart: The chart visually represents the initial and final positions of the segment AB and the location of the CoR.
6. Reset: Click “Reset” to return to the default values.
7. Copy: Click “Copy Results” to copy the main result and intermediate values to your clipboard.

This Center of Rotation (CoR) calculation tool simplifies finding the pivot point.

Key Factors That Affect Center of Rotation (CoR) calculation Results

• Marker Placement Accuracy: Small errors in identifying marker coordinates can lead to significant changes in the calculated CoR, especially if the angle of rotation is small or the markers are close together.
• Rigid Body Assumption: The method assumes the distance between markers A and B remains constant. If the body deforms, or markers are on soft tissue that moves relative to the bone, the Center of Rotation (CoR) calculation will be less accurate. Check our guide on biomechanics calculator principles.
• Planar Motion Assumption: This 2D CoR calculation assumes movement occurs purely in the XY plane. Out-of-plane motion will introduce errors. For 3D motion, a helical axis of motion is more appropriate.
• Magnitude of Rotation: The CoR calculated is for the specific displacement from position 1 to 2. For larger rotations, the average CoR might differ from the instantaneous center of rotation at different points in the motion.
• Distance Between Markers: Markers that are further apart generally yield a more stable and accurate CoR calculation than markers that are very close together.
• Data Smoothing: If coordinates are derived from noisy motion capture data, smoothing the marker trajectories before applying the Center of Rotation (CoR) calculation can improve results.
• Time Interval/Displacement Size: The CoR represents the average center for the movement between the two instances. A smaller interval approximates the instantaneous center of rotation better.

Understanding these factors is crucial for an accurate Center of Rotation (CoR) calculation.

Frequently Asked Questions (FAQ)

What is the difference between Center of Rotation (CoR) and Instantaneous Center of Rotation (ICR)?

The CoR calculated here is for a finite displacement between two positions. The ICR is the center of rotation at a specific instant in time during continuous motion. If the two positions are very close (small time interval), the CoR approximates the ICR.

What if the perpendicular bisectors are parallel?

If the perpendicular bisectors are parallel and distinct, there is no finite CoR, implying pure translation (or an error). If they are coincident, there are infinite CoRs (pure rotation around any point on that line, which is unlikely for two distinct markers unless they swap positions in a very specific way or there’s an error).

Can I use this for 3D motion?

No, this calculator is specifically for 2D (planar) motion. 3D rotation occurs around an axis, not a point, and is described by a helical axis of motion.

What units should I use?

You can use any consistent unit of length (mm, cm, m, inches) for all coordinates. The CoR coordinates will be in the same units.

How does the CoR relate to joint centers?

For many biological joints, the CoR provides an estimate of the functional joint center during a specific movement. However, biological joints often have migrating CoRs. More about the CoR formula and its applications can be found here.

What if the body isn’t perfectly rigid?

If there’s relative movement between the markers due to soft tissue or deformation, the calculated CoR will be an approximation and may vary depending on the markers chosen. Strive to place markers on areas with minimal soft tissue movement relative to the underlying bone to improve the rigid body assumption for Center of Rotation (CoR) calculation.

Why is the CoR sometimes far from the physical joint?

This can happen if the movement involves a combination of rotation and translation, or if there are errors in marker data. It could also indicate that the joint’s true axis of rotation is complex or lies outside the joint structure itself for that particular motion.

How small should the movement be between the two positions?

For approximating the ICR, the movement should be as small as possible while still being accurately measurable. For a finite rotation CoR, any distinct positions can be used, but the CoR represents the average over that displacement. Understanding how to find center of rotation accurately is key.