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Start with coordinate systems. I've drawn one example. In my analysis, if all $q_i = 0$ then the manipulator would point straight up. You can choose other coordinate frames to get the same result. Build your rotation matrices from the coordinate systems you set up. The rotation matrix from coordinate system $0$ to coordinate system $1$ is $$_0^1R = \... 5 To extends the answer from the_parzival a bit: There are different kind of robots so that 'robot state' can have different meanings. If you have a drone or Roomba-robot, the most important state is related to its pose (position, orientation, speed, acceleration, ...). Other states are the Battery state, Motor speeds (and temperatures), essentially every ... 5 If the drone is not falling (holding height in the sky), and it's not accelerating in any particular direction, then the accelerometer should be reading:$$ a = \left[ \begin{array}{} g_x \\ g_y \\ g_z \end{array}\right] $$where g_N is the component of gravity along each axis. If the drone is upright and stationary, and the accelerometer is oriented ... 4 In your forward kinematics transformation matrix (4x4, incl. also translation, of just 3x3) the orientation of the end-effector is expressed relative to the base (or world) coordinate system. SteveO described very well how to obtain the 3x3 rotation matrix, similarly you can obtain also the 4x4 transformation matrix If you have the matrix you have to ... 4 The rotation error between two frames can be viewed in two ways: The orientation of one frame as seen from the other, calculated by multiplying the inverse of the observing frame by the observed frame. For frames E and H, this error in your notation would be$$ \tag{1} {R_{1}}^{E}_{H} = (R^{W}_{E})^{-1}R^{W}_{H} = R^{E}_{W}R^{W}_{H} $$for H as seen ... 3 There is no way that you can solve for an IK solution without you -- either explicitly or implicitly -- specifying a criterion for choosing one solution among many others. But that does not mean that solving an IK problem without having any preference on the orientation (or translation) is not possible. If you solve IK problems analytically, you can just ... 3 Up till now, I am using the Euler angle representation but I have not been successful. After referring to the book "Robotics Modelling, Planning and control" by Bruno Siciliano form Springer publication. It is concluded that the Euler angle is only used for the manipulator having spherical wrist; the robot which doesn't have one need to use another ... 3 I encountered the same puzzle. I had a clue at the beginning that the gravity information is contained within accelerometer measurements due to aerodynamic drag. Then I found a paper The True Role of Accelerometer Feedback in Quadrotor Control, which had proved the idea. My simulation also revealed the process of stabilization. As long as a quadcopter ... 3 Accelerometers measure kinematic acceleration with the addition of gravity. So for an accel to measure 0, the vehicle would need to be accelerating downward at g. To get inertial acceleration out of an accel measurement one simply needs to subtract the acceleration measured by the IMU when the IMU is static. So, assuming the coordinate system of the ... 3 Calibration procedures for magnetometers exist, to compensate for soft iron (nearby ferromagnetic objects) and hard iron (nearby magnetic fields) offsets, which skew the measurements. However, these procedures usually map a static disturbance correction and apply it to all new measurements. On the contrary, your environment changes from one end of the tube ... 3 As Brian indicated in a comment, you simply need to convert your rotation matrix (or Euler angles) into a quaternion. Maths - Conversion Matrix to Quaternion is my favorite site for geometric conversions. Quaternions are a great representation and have a number of benefits over other representations, so you should definitely read up on them. 3 A rotation matrix represents the rotation between two frames. Therefore, it does not make sense to talk about in which "one" frame the error rotation is expressed. Namely, the rotation matrix R^B_A represents the rotation from frame B to frameA. Therefore, the error rotation matrix you defined as$$ R_{error} = R^W_E R^H_W $$can also be seen as the ... 2 I'm not familiar with your device, but here are a few tips that might help... Since the device is using a quaternion to describe an orientation, they must also specify some base coordinate frame against which that rotation applies. The w term in a quaternion is the cosine of the half-angle of desired rotation around the (x,y,z) vector component. If you don'... 2 Its a bit difficult to give you an exact answer because its quite an open question. You are sorting the bricks based on an optical characteristics so I can't see any way of doing it other than by using an optical sensor. If you start sorting them by magnetic properties then you aren't sorting them by their colour anymore. So what is the reason you don't ... 2 Your own equations make a lot of assumptions about the problem you are trying to solve. They are sufficient to solve your problem, but will not work for certain conditions (z = 0 for instance). Euler angles are similar. A way describe orientation (we typically use orientation to mean all angles, not just pitch and roll as you are using here). Euler angles ... 2 You say, Then why can't we track [the angles] for first ten seconds and then keep subtracting the present angles from initially calculated angles for every ten seconds during movement? You can't subtract angles from a gyroscope reading because the gyroscope measures angular velocity. This is kind of like asking why you can't subtract your odometer ... 2 1) Yes, a megnetometer is an instrument that measures magnetism — either the magnetization of a magnetic material like a ferromagnet, or the direction, strength, or relative change of a magnetic field at a particular location. I imagine the GPS sensor gives some approximate latitude and longitude - this isn't something a magnetometer could give you. 2) As ... 1 The state of the Robot refers to the properties of the robot that you want to estimate. If you take a 2D robot as an example, then you might be interested in its 2d position which refers to$$ X(t) = \begin{bmatrix} x(t) \\ y(t) \\ \end{bmatrix} $$If you also want to estimate its velocity, then you can add another element to its state vector and ... 1 It may be helpful to not get caught up in the author’s description of frames with respect to these vectors. The author is stating that an axial vector, like a force vector, acts along a line. But a polar vector, like a torque, acts about an axis. 1 A rotation matrix is also called a director cosine matrix. The elements of the rotation matrix are the cosines of the unit vectors of two coordinate systems involved. You can find a more generic explanation here. Let \angle (e_{2,i}, e_{e,j}) denote the angle between the angle between unit vector on the i axis of the fixed reference frame and the unit ... 1 It looks like both frames are attached to the end effector at point P, and are offset by a constant angle \beta. So you don't really have to consider the D-H parameters to answer this question - just do a simple rotation matrix evaluation. So how are the frames, F_2 = [\hat{x_2} \; \hat{y_2} \; \hat{z_2}]^T and F_e = [\hat{x_e} \; \hat{y_e} \; \hat{... 1 Your A_{0.3} matrix is a 4x4 transformation matrix. In a general form these 4x4 matrices can be subdivided into a 3x3 rotation and a 3x1 translation part. (The remaining parts can be used for scaling, but are not used in robotics)  A_{0,3}= T_{4 \times 4}= \begin{pmatrix} R_{3 \times 3} & T_{3 \times 1}\\ 0_{1 \times 3} & 1 \end{pmatrix}  The ... 1 Sim3 is just the lie group associated a similarity transform which is the exact same thing as your 3D Affine transform. If you work out the multiplication for your two matrices you would find that they end up with the same result as we are dealing with homogenous coordinates. Therefore the following is true$$x=kx$$where k is just some constant, and x ... 1 I think you'll find section IV. Loop Closure useful here: http://www.roboticsproceedings.org/rss06/p10.pdf 1 The answer by parzival is only a partial answer. Especially with robots with more than 3 axes, the matrix cannot be solved for a deterministic state, a famous example of this is Dirac's belt trick, where with only the 3 DOF that a human arm and shoulder provide, you cannot have any deterministic state using the matrix provided above. The true state of a ... 1 The same as for positions, the angular components of \dot x are angular velocities. Expressing them as motion difference over time is generally correct. The unit of measurment in this case needs to be rad/s. The other problem can come from the how the rotations are expressed. There are many conventions on how rotations can be expressed and Euler angles ... 1 Usually the magnetometer is used to find the yaw. It acts as a digital compass in this case. To calculate roll and pitch you need an accelerometer. But there are some techniques that can be used to calculate the roll and pitch using the magnetometer. For that you need to place a magnet close to the mobile phone and observe the sensor values. Using these ... 1 I recommend using april tags. They are similar to qr codes but were specifically designed with robotics in mind. There is a library on the april tag website. The library can be used to solve for the position and orientation of each april tag in a given image. https://april.eecs.umich.edu/software/apriltag/ 1 General movement is not a problem even at high velocity but, as you already described, acceleration is. There are a couple mitigating factors that allow many platforms to still (mostly) assume the accelerometer is a gravity measurement Short bursts of acceleration can be filtered out (e.g. if ||a|| >> ||a_g|| ignore the reading) High frequency ... 1 Referring to the equation \dot{x} = J(q)\dot{q}, what the last three elements of \dot{x} should be depends on what kind of Jacobian you are using. Geometric Jacobian: Suppose your Jacobian is computed as$$J = \begin{bmatrix}J_1 & J_2 & \cdots J_n\end{bmatrix},$$where$$J_i = \begin{cases} \begin{bmatrix}z_{i - 1}\\0_{3\times1}\end{bmatrix} ...