Homework 3

Course	Introduction to Robotics
Course Number	00350001

סטודנט א’	סטודנט ב’	סטודנט ג’
עידו פנג בנטוב	ניר קרל	אופיר רובין
CLASSIFIED	CLASSIFIED	CLASSIFIED
CLASSIFIED	CLASSIFIED	CLASSIFIED

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Question 1

Part a

From previous homework:

Figure HW3.1: Assigning all the frames according to the convention.

The D–H Parameters are:

Which gives us the following forward kinematics:

Given , and we want to move the robot so that:

We already know that , where is a given parameter. Therefore, and .

From (HW3.1), we can find :

From (HW3.2):

Therefore:

From (HW3.3):

Part b

We now know that:

If the tool’s tip is located at in gripper frame coordinates, then the total transformation from world frame coordinates is:

Therefore, when the tool tip is located at , then:

We already know that .

Therefore:

From (HW3.4):

Therefore:

From (HW3.5):

Since , we must have .

Therefore:

From (HW3.6):

Since , we have .

This gives us two solutions:

Figure HW3.2: (Left) Solution where . (Right) Solution where .

This manipulator has 2 solutions:

• 2 solutions for (elbow-up/elbow-down configurations)

graph TD
	A["$$({p}_{x}, {p}_{y}, {p}_{z})$$"] --> B["$${\theta}_{3} = 90° + \alpha$$"]
	B --> C["$${d}_{4} = \frac{{L}_{1}-{p}_{x}}{-\sin\alpha}$$"]
	C --> E["$${\theta}_{2} = \cos^{-1}\left(\frac{{p}_{y}}{\cos\alpha \cdot {d}_{4}+{L}_{2}}\right)$$"]
	C --> F["$${\theta}_{2} = 2\pi - \cos^{-1}\left(\frac{{p}_{y}}{\cos\alpha \cdot {d}_{4}+{L}_{2}}\right)$$"]
	E --> G["$${d}_{1} = {p}_{z}-{s}_{2}(\cos\alpha \cdot {d}_{4}+{L}_{2})$$"]
	F --> H["$${d}_{1} = {p}_{z}-{s}_{2}(\cos\alpha \cdot {d}_{4}+{L}_{2})$$"]

Question 2

From previous homework:

Figure HW3.3: Assigning all the frames according to the convention.

The D–H Parameters are:

Which gives us the following forward kinematics:

Where:

Given , and we want to move the robot so that:

From (HW3.9) we can easily see that:

To find the remaining joint variables, we need to solve equations (HW3.7) and (HW3.8) for and .

From (HW3.7) and (HW3.8):

Squaring both equations and adding them:

Therefore:

For , we can use:

Using for proper quadrant handling:

This manipulator has 2 solutions:

• 2 solutions from the sign in (forward/backward prismatic motion)

graph TD
	A["$$({p}_{x}, {p}_{y}, {p}_{z})$$"] --> B["$${d}_{1} = {p}_{z} - {L}_{4}$$"]
	B --> C["$${d}_{3} = \pm\sqrt{({p}_{x} - {L}_{1}{c}_{{\phi}_{1}})^{2} + ({p}_{y} - {L}_{1}{s}_{{\phi}_{1}})^{2}}$$"]
	C --> D["$${d}_{3} = +\sqrt{\cdots}$$"]
	C --> E["$${d}_{3} = -\sqrt{\cdots}$$"]
	D --> F["$${\theta}_{2} = \mathrm{atan2}(\cdots) - {\phi}_{1}$$"]
	E --> G["$${\theta}_{2} = \mathrm{atan2}(\cdots) - {\phi}_{1}$$"]

Question 3

From previous homework:

Figure HW3.4: Assigning all the frames according to the convention.

The D–H Parameters are:

Which gives us the following forward kinematics:

Given , and we want to move the robot so that:

We are also given that:

From equations (HW3.10) and (HW3.11), we can solve for :

Dividing (HW3.11) by (HW3.10):

Therefore:

Note that can also be (pointing in the opposite direction), giving us a second solution branch.

Now, substituting back into equation (HW3.10):

Let’s denote:

So we have:

From equation (HW3.12):

Using the constraint , we have .

From equations (HW3.14) and (HW3.15):

Rearranging:

Let:

Then:

Squaring and adding:

Therefore:

The gives us the elbow-up and elbow-down configurations.

For , we can use:

Expanding the trigonometric terms:

This system can be written in the form:

where and .

We can express this system as a rotation matrix:

For any vector rotated by angle to give :

Applying this identity to our system where is rotated by :

Therefore:

Finally, from the constraint equation (HW3.13):

Total: 4 distinct solutions (2 × 2 = 4 combinations)

flowchart TD

	A["$$({p}_{x}, {p}_{y}, {p}_{z})$$"] --> B["$${\theta}_{1} = \tan^{-1}({p}_{y}, {p}_{x})$$"]

	B --> D["$${\theta}_{1}$$"] & E["$${\theta}_{1} + \pi$$"]

	D --> H["$${\theta}_{3} = +\cos^{-1}(\cdots)$$"] & I["$${\theta}_{3} = -\cos^{-1}(...)$$"]

	E --> J["$${\theta}_{3} = +\cos^{-1}(\cdots)$$"] & K["$${\theta}_{3} = -\cos^{-1}(\cdots)$$"]

	H --> L["$${\theta}_{2} = \mathrm{atan2}(\cdots) - \mathrm{atan2}(\cdots)$$"]
	I --> M["$${\theta}_{2} = \mathrm{atan2}(\cdots) - \mathrm{atan2}(\cdots)$$"]
	J --> N["$${\theta}_{2} = \mathrm{atan2}(\cdots) - \mathrm{atan2}(\cdots)$$"]
	K --> O["$${\theta}_{2} = \mathrm{atan2}(\cdots) - \mathrm{atan2}(\cdots)$$"]

	L --> P["$${\theta}_{4} = \alpha - {\theta}_{2} - {\theta}_{3}$$"]
	M --> Q["$${\theta}_{4} = \alpha - {\theta}_{2} - {\theta}_{3}$$"]
	N --> R["$${\theta}_{4} = \alpha - {\theta}_{2} - {\theta}_{3}$$"]
	O --> S["$${\theta}_{4} = \alpha - {\theta}_{2} - {\theta}_{3}$$"]

Notes:

• The solution exists only if
• Each solution corresponds to a different robot configuration: (shoulder-left/right) × (elbow-up/down)