# 3 degrees of freedom analytical solution

I have got a robot that exactly looks like as shown in the figure above. I have worked out the inverse kinematics analytical solution without the base rotation (considering 2 dof alone) but I am not able to find the analytical solution including the base(3 dof). How do I find the anlytical solution for this robot ??

You can find many examples for this type of robot arm online. It is kinematically similar to the first three joints of a Puma 560, which was used extensively as an example when robot kinematics and controls algorithms were being developed in academia.

Try section 4.2 of this paper:

http://deepblue.lib.umich.edu/bitstream/handle/2027.42/6192/bac6709.0001.001.pdf

• Hey thank you so much ! I did go through the PDF, I just have a doubt on that, in page 13 where it says OA = d2, is OA the pertubations from the side of the base to the joint ?? – Bhuvanesh Narayanan Dec 22 '15 at 0:52
• It is the $d_2$ parameter from the figure on page 5. – SteveO Dec 22 '15 at 0:55
• on your image d2=0 – 50k4 Dec 22 '15 at 7:58
• Ahh now I get it !! Just a final doubt, the theta1 shown in the figure on page 13, I am trying to understand why that particular point has been taken as theta1, I don't understand how that theta1 makes a reference with the arms position. The ARM has ben indicated by bold lines whereas there is a additional line ox1 which makes the theta 1. Why could it be so ? – Bhuvanesh Narayanan Dec 22 '15 at 11:10
• Correct. You also need to handle the basics, such as making sure you transition smoothly when the $0$, $\pi$, or $2 \pi$ boundaries are crossed (you don't want your motor to try to spin -359.9 degrees when you cross one of those values). You will need to choose between an "elbow up" or an "elbow down" configuration as well. I believe that is described pretty well in that paper. – SteveO Dec 22 '15 at 13:36

Your manipulator is almost identical to the Phantom Omni in the below picture,

In this paper Teleoperation with inverse dynamics control for PHANToM Omni haptic device, the Kinematics and Dynamics of the device are provided.

For fun, I've simulated the model in the aforementioned paper. I've chosen the PD controller to carry out the simulation. For the desired trajectories, I've chosen the following

$$\theta^{d}_{1}(t) = 0.1\sin(\pi t) \\ \theta^{d}_{2}(t) = 0.1\cos(\pi t) \\ \theta^{d}_{2}(t) = 0.1\cos(\pi t) \\$$

The controller input is

$$u = M(q)*( \ddot{q}^{d} + K_{d} \dot{e}(t) + K_{p}e(t) ) + V(q,\dot{q}) \dot{q} + N$$

The result is

If you have some uncertianties in your model, you should choose PID controller instead of the PD controller.

• hey thank you so so much, I am almost there ! I need the geometric solution because using this I can create an FPGA design to find the theta's ! I checked the link of ur paper, I hope I get this somehow, In page 2122 I can see how they find the 3 angles, but I do not undrstand L3 and L4. Like how can I calculate these values ??I know that L1 and L2 are the length of the robot but L3 and L4 are workspace transformation offset & from the figure I see tht it changes fr evry postion.How do I calculate this ? Sorry I am from a pure electronics background ! would be great if I find the way for this – Bhuvanesh Narayanan Dec 24 '15 at 21:14
• There is a picture in the paper which shows L3, L4. You need just to measure them if you have the actual manipulator. – CroCo Dec 24 '15 at 21:19
• Yes I see the diagram but it says that 'L3 and L4 are the workspace transformation offsets between the origin of the end effectors and the first joint' bu the end effector position keeps changing depending on the new point it is at is'nt it ? Mine looks more like this bestperformancegroup.com/wp-content/uploads/2013/10/… So I just measure from the origin of the base to the origin of the end effector just like for the phantom omni then that shud be my L4 and similarly my L3 keeping the robot in at its rest position ?? – Bhuvanesh Narayanan Dec 24 '15 at 21:29
• Any clue mate on this one ? If I keep the end effector at the origin position as shown in the pdf in page 2122 then measure the L3 and L4 as shown, it would work right ? – Bhuvanesh Narayanan Dec 25 '15 at 11:24