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Robot control part 2: Jacobians, velocity, and force


Jacobian matrices are a super useful tool, and heavily used throughout robotics and control theory. Basically, a Jacobian defines the dynamic relationship between two different representations of a system. For example, if we have a 2-link robotic arm, there are two obvious ways to describe its current position: 1) the end-effector position and orientation (which we will denote $latex \textbf{x}$), and 2) as the set of joint angles (which we will denote $latex \textbf{q}$). The Jacobian for this system relates how movement of the elements of $latex \textbf{q}$ causes movement of the elements of $latex \textbf{x}$. You can think of a Jacobian as a transform matrix for velocity.

Formally, a Jacobian is a set of partial differential equations:

$latex \textbf{J} = \frac{\partial \textbf{x}}{\partial \textbf{q}}$.

With a bit of manipulation we can get a neat result:

$latex \textbf{J} = \frac{\partial \textbf{x}}{\partial t} \; \frac{\partial t}{\partial \textbf{q}} \rightarrow \frac{\partial \textbf{x}}{\partial \textbf{t}} =…

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[edu]Harmonic drive

What is a harmonic drive?

A cool kind of gear.  It was invented in 1957 by C.W. Musser.

220px-Harmonic_drive_animationThe advantages include: no backlash, compactness and light weight, high gear ratios, reconfigurable ratios within a standard housing, good resolution and excellent repeatability (linear representation) when repositioning inertial loads, high torque capability, and coaxial input and output shafts.High gear reduction ratios are possible in a small volume (a ratio from 30:1 up to 320:1 is possible in the same space in which planetary gears typically only produce a 10:1 ratio).

Disadvantages include a tendency for ‘wind-up’ (a torsional spring rate) and potential degradation over time from mechanical shocks and environment.



Cross-section of a strain wave gearing mechanism.220px-Harmonic-drive-explanationA: circular spline (fixed)
B: flex spline (attached to output shaft, not shown)
C: wave generator (attached to input shaft, not shown)


For a strain wave gearing mechanism, the gearing reduction ratio can be calculated from the number of teeth on each gear:

\text{reduction ratio} = \frac {\text{flex spline teeth} - \text{circular spline teeth}} {\text{flex spline teeth}}

For example, if there are 202 teeth on the circular spline and 200 on the flex spline, the reduction ratio is (200 − 202)/200 = −0.01

Thus the flex spline spins at 1/100 the speed of the wave generator plug and in the opposite direction. This allows different reduction ratios to be set without changing the mechanism’s shape, increasing its weight, or adding stages.

Robot control part 1: Forward transformation matrices


I’m doing a tour of learning down at the Brains in Silicon lab run by Dr. Kwabena Boahen for the next month or so working on learning a bunch about building and controlling robots and some other things, and one of the super interesting things that I’m reading about is effective methods for force control of robots. I’ve mentioned operational space (or task space) control of robotic systems before, in the context of learning the inverse kinematic transformation, but down here the approach is to analytically derive the dynamics of the system (as opposed to learning them) and use these to explicitly calculate control signals to move in task space that take advantage of the passive dynamics of the system.

In case you don’t remember what those words mean, operational (or task) space refers to a different configuration space than the basic / default robot configuration space. FOR EXAMPLE:…

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[edu] Be Flexible

Here is a project of TU Delft – MINDWALKER exoskeleton is equipped with powerful and efficient actuators to support hip adduction/abduction, flexion/extension, and knee flexion/extension. The actuators are designed using Series Elastic Actuation (SEA) principle. The exoskeleton has also integrated advanced sensors such as precise joint angle sensors, Inertial Measurement Units (IMUs), force/torque sensors, obstacle detection sensors and etc. All the actuators and sensors are connected to a network created by Ethernet-based fast fieldbus system called EtherCAT.

Video MindWalker

And some publications:

Modeling, Design, and Optimization of Mindwalker Series Elastic Joint

Actively Controlled Lateral Gait Assistance in a Lower Limb Exoskeleton


[edu]Keep the balance

Here are some useful stuff to read on the subject of balance (no Dzen Budhisum implemented anyhow). With plenty of formulas for the once who enjoy extreme experiences.

Balancing Control of AIT Leg Exoskeleton Using ZMP based FLC – a study by Narong Aphiratsakun and Manukid Parnichkun, School of Engineering and Technology, Asian