Unit 5: Inverse Kinematics

ECE245 — Elements Of Robotics Laboratory 9 min read

I. Orientation: Inverse Kinematics

Inverse kinematics (IK) determines the joint variables required to place a robot’s end-effector at a specified position and orientation. For a two-axis planar articulated robot, the problem is to calculate the two joint angles from a desired point ((x,y)), using the known link lengths (a_1) and (a_2). The same physical relationship can be expressed geometrically or systematically through the Denavit-Hartenberg (DH) algorithm.

Governing principle: The desired Cartesian position is converted into joint coordinates by solving the robot’s forward-kinematics equations in reverse.
Robot structure: A planar two-axis arm has two revolute joints, commonly called the shoulder or base joint (\theta_1) and elbow joint (\theta_2).
Coordinate convention: The base is placed at the origin, the (x)-axis is horizontal, and the (y)-axis is vertical. Positive joint angles are normally measured counterclockwise.
Known quantities: Link lengths (a_1) and (a_2), target coordinates (x) and (y), and the permitted joint-angle ranges.
Unknown quantities: The joint angles (\theta_1) and (\theta_2).
Multiple solutions: A reachable point generally has two configurations: elbow-up and elbow-down.
Reachability condition: The target must satisfy
[
|a_1-a_2| \leq r \leq a_1+a_2,
]
where (r=\sqrt{x^2+y^2}) is the distance from the base to the target.
Practical convention: Real robot commands may use servo offsets, reversed directions, and degree units even though the mathematical derivation uses radians.

II. Geometric Approach to the Planar Arm

A. Inverse kinematics of two-axis planar articulated robot using geometric approach

The geometric approach treats the two links and the target point as a triangle, allowing the joint angles to be obtained from the cosine rule and basic trigonometry.

Target distance: The straight-line distance from the base to the target is
[
r=\sqrt{x^2+y^2}.
]
Here, (x) and (y) are target coordinates in the base frame, and (r) is measured in the same length unit as (a_1) and (a_2).
Triangle formation: The links (a_1) and (a_2), together with the line from the base to ((x,y)), form a triangle. Its side lengths are (a_1), (a_2), and (r).
Elbow angle: Applying the cosine rule gives
[
\cos\theta_2=
\frac{x^2+y^2-a_1^2-a_2^2}{2a_1a_2}.
]
Therefore,
[
\theta_2=\operatorname{atan2}\left(\pm\sqrt{1-\cos^2\theta_2},\cos\theta_2\right).
]
The plus and minus signs produce the two elbow configurations.
Shoulder angle: The angle from the positive (x)-axis to the target is
[
\alpha=\operatorname{atan2}(y,x).
]
The angle between the first link and the target line is
[
\beta=\operatorname{atan2}\left(a_2\sin\theta_2,\ a_1+a_2\cos\theta_2\right).
]
Hence,
[
\theta_1=\alpha-\beta.
]
Meaning of atan2: The two-argument function (\operatorname{atan2}(y,x)) preserves the correct quadrant. Using (\tan^{-1}(y/x)) alone can produce an incorrect shoulder angle when (x) is negative.

Complete geometric solution:

TEXT

  r2 = x*x + y*y
  c2 = (r2 - a1*a1 - a2*a2) / (2*a1*a2)
  s2 = +/- sqrt(1 - c2*c2)
  theta2 = atan2(s2, c2)
  theta1 = atan2(y, x) - atan2(a2*s2, a1 + a2*c2)

Here, c2 and s2 are (\cos\theta_2) and (\sin\theta_2). The + value gives one elbow posture and the - value gives the other.

Worked example: For (a_1=10\text{ cm}), (a_2=8\text{ cm}), and target ((x,y)=(12,6)\text{ cm}),
[
r^2=12^2+6^2=180,
]
[
\cos\theta_2=\frac{180-100-64}{160}=0.1.
]
Thus, (\theta_2\approx\pm84.3^\circ). For the positive-sine solution,
[
\theta_1=\operatorname{atan2}(6,12)-
\operatorname{atan2}(8\sin84.3^\circ,10+8\cos84.3^\circ),
]
giving approximately (\theta_1= -7.1^\circ). The negative-sine solution produces the corresponding elbow-down posture.

B. Reachability and solution selection

The geometric solution is useful only when the target lies inside the arm’s annular workspace and the resulting angles are acceptable for the hardware.

Outer boundary: The maximum reach is (a_1+a_2), occurring when both links are fully extended.
Inner boundary: If (a_1\neq a_2), the minimum reach is (|a_1-a_2|), occurring when the links fold over one another.
Unreachable target: If
[
\left|\frac{x^2+y^2-a_1^2-a_2^2}{2a_1a_2}\right|>1,
]
no real elbow angle exists.
Numerical tolerance: Due to sensor or floating-point error, a value such as (1.000001) may be clamped to (1), but a substantially larger value indicates an invalid target.
Configuration choice: The controller selects elbow-up or elbow-down according to joint limits, obstacle clearance, and the previous arm position.
Singular posture: When (\sin\theta_2=0), the arm is fully stretched or folded. Small Cartesian movements can then require large changes in joint angles.

III. DH Algorithm for Inverse Kinematics

A. DH algorithm

The Denavit-Hartenberg algorithm represents a serial robot as a sequence of homogeneous transformations, providing a systematic route from joint variables to the end-effector pose and from that pose to inverse-kinematic equations.

Purpose: DH notation standardizes the placement of coordinate frames on robot links so that every joint transformation uses four parameters.
DH parameters: For link (i), the parameters are:
- (a_i): link length, measured along (x_i).
- (\alpha_i): link twist, the rotation about (x_i).
- (di): link offset, measured along (z{i-1}).
- (\thetai): joint angle, the rotation about (z{i-1}).
Standard DH transformation: The transformation from frame (i-1) to frame (i) is
[
{}^{i-1}T_i=
R_z(\theta_i)T_z(d_i)T_x(a_i)R_x(\alpha_i).
]
In matrix form,
[
{}^{i-1}T_i=
\begin{bmatrix}
c_i&-si c{\alpha i}&si s{\alpha i}&a_i c_i\
s_i&ci c{\alpha i}&-ci s{\alpha i}&a_i si\
0&s{\alpha i}&c_{\alpha i}&d_i\
0&0&0&1
\end{bmatrix},
]
where (c_i=\cos\theta_i), (s_i=\sin\thetai), (c{\alpha i}=\cos\alphai), and (s{\alpha i}=\sin\alpha_i).
Planar simplification: For two revolute axes perpendicular to the (x)-(y) plane, (d_1=d_2=0) and (\alpha_1=\alpha_2=0). The transformations reduce to
[
{}^0T_1=R_z(\theta_1)T_x(a_1),\qquad
{}^1T_2=R_z(\theta_2)T_x(a_2).
]
Forward position from DH multiplication:
[
{}^0T_2=
\begin{bmatrix}
\cos(\theta_1+\theta_2)&-\sin(\theta_1+\theta_2)&0&
a_1\cos\theta_1+a_2\cos(\theta_1+\theta_2)\
\sin(\theta_1+\theta_2)&\cos(\theta_1+\theta_2)&0&
a_1\sin\theta_1+a_2\sin(\theta_1+\theta_2)\
0&0&1&0\
0&0&0&1
\end{bmatrix}.
]
The final column gives the end-effector position.
Inverse step: Equating the final-column entries to the target gives
[
x=a_1\cos\theta_1+a_2\cos(\theta_1+\theta_2),
]
[
y=a_1\sin\theta_1+a_2\sin(\theta_1+\theta_2).
]
Eliminating (\theta_1) produces the same cosine-rule expression for (\theta_2), followed by the atan2 expression for (\theta_1).

B. DH algorithm with planar joint conventions

Correct frame assignment is essential because an incorrect axis or link-length convention changes the resulting equations even when the algebra is performed correctly.

Joint axes: For a planar revolute arm, each (z)-axis is perpendicular to the plane of motion and passes through its associated joint.
Frame origins: Frame ({0}) is attached to the base joint; frame ({1}) is attached to the elbow joint; frame ({2}) is attached to the end-effector or wrist reference point.
Link parameters: With standard DH placement, the two physical link lengths appear as (a_1) and (a_2). Their measured values must include the actual distance between joint axes.
Angle zero: The mathematical zero angle must be related to the physical servo zero. A calibration offset (\deltai) may be required:
[
\theta{i,\text{command}}=\theta_{i,\text{mathematical}}+\delta_i.
]
Orientation information: A two-axis planar arm controls (x), (y), and the tool orientation (\phi=\theta_1+\theta_2), but only two independent joint variables are available. Therefore, a complete pose cannot generally be prescribed arbitrarily; position or position plus a dependent orientation must be specified.
Verification: Substitute the calculated angles into the DH forward equations. The position error is
[
e=\sqrt{(x-x_d)^2+(y-y_d)^2},
]
where ((x_d,y_d)) is the desired point and ((x,y)) is the computed point.

IV. Orangewood Robotic Arm

A. Orangewood Robotic Arm

The Orangewood Robotic Arm provides a physical platform for implementing and validating inverse kinematics by mapping calculated joint angles to motor commands and comparing the resulting end-effector position with the desired coordinates.

Laboratory purpose: The arm demonstrates the complete chain: measure geometry, define a target, calculate IK, command joints, and observe positioning error.
Model selection: For the two-axis planar experiment, isolate the shoulder and elbow revolute joints. Keep the base orientation, gripper orientation, and any out-of-plane motion fixed.
Required measurements: Measure (a_1) from the shoulder-axis center to the elbow-axis center and (a_2) from the elbow-axis center to the selected end-effector reference point.
Coordinate calibration: Establish the base origin and record the mathematical zero positions. A camera, ruler, or marked workspace can be used to identify ((x,y)) in centimetres or millimetres.
Command conversion: Convert radians to degrees before sending commands if the arm interface expects degrees:
[
\theta{\deg}=\theta{\rad}\frac{180}{\pi}.
]
Apply servo offsets and direction reversals after calculating the mathematical angles.
Safe operation: Test targets inside the reachable workspace and within the manufacturer’s joint limits. Begin with low-speed motion and keep the gripper clear of obstructions.

Practical algorithm: