Stuart O Anderson

12-10-2007, 03:38 AM

> I don't pretend to understand the math of the "jacobian" could you explain

> the concept a little more.

A jacobian is the matrix of partial derivatives of a vector valued

function. In this case we mean the partials of the forward kinematics

function that gives the Cartesian position of the foot as a function

of joint angles. So the partial derivatives of this function give the

relationship between joint velocities and velocities of the foot in

Cartesian coordinates. The jacobian matrix contains an element for

each pair of foot position coordinate and joint angle [i.e. it is a

NxM matrix where you have N joints and M coordinates describing the

position and rotation of the foot].

If you hold your arm out straight in front of you with your palm

facing the ground and the axis of your elbow joint vertical you can

figure out a few elements of the kinematic jacobian of your hand

intuitively. Bending your elbow slightly moves your hand medially at

a rate equal to the length of your forearm times the velocity with

which you move your elbow joint. Rotating your shoulder about a

vertical axis at the same rate moves your hand in the same direction,

but about twice as fast, because your arm is about twice as long as

your forearm. Now, to get at the reason for having bent knees during

gait, try to move your hand directly towards your shoulder - doing

this requires moving the elbow and shoulder simultaneously in opposite

directions. If you move your hand at a constant velocity towards your

shoulder you'll see that the angular rate that you move your elbow at

slows down as your get closer to your shoulder. As you move further

from your shoulder, and closer to the 'kinematic singularity' that

occurs when you have a straight arm, the joint velocities needed to

keep your hand moving at a constant velocity increases. Analytically

there is an infinite vertical asymptote caused by dividing by a sin(

theta ) term as you approach the singular configuration - requiring

infinite joint velocity to maintain the same finite hand velocity.

> Would you say that the straight knee approach is another example of

> simplifying the human gait model. This model uses the momentum of the CoM

> Plus a small added energy input, in the robot case from electric motors,

> which is much more energy efficient than the Asimo model.

I think that's a fair assessment. There are unpowered passive dynamic

walkers (or more properly gravity powered passive dynamic walkers)

that have no motors or sensors and walk down slight inclines to

compensate for energy loss due to foot impact.

> Do you think a combination of the Asimo and the rocker foot robot would be a

> possibility that would be more analogous to 'normal or usual' human gait. EG

> it could recognise rough terrain or obstacles and converts from the

> efficient stiff knee rocker foot gait to the bent knee stable posture

> progression at the appropriate time.

There is definitely interest in merging the insights provided by these

two models. I think the key difference between them is that ASIMO and

kin have a stability metric that can be evaluated and enforced at

every instant in time, while the passive dynamic walkers use foot

placement and the dynamics of foot impact to stabilize and reject

disturbances over multiple steps. Your comments on energy storage in

human muscles and tendons remind me of the work being done by Hurst

and Grizzle on AMASC actuator based legs

[http://www.biped.ri.cmu.edu/] - that can efficiently store and return

mechanical energy.

Stuart

> the concept a little more.

A jacobian is the matrix of partial derivatives of a vector valued

function. In this case we mean the partials of the forward kinematics

function that gives the Cartesian position of the foot as a function

of joint angles. So the partial derivatives of this function give the

relationship between joint velocities and velocities of the foot in

Cartesian coordinates. The jacobian matrix contains an element for

each pair of foot position coordinate and joint angle [i.e. it is a

NxM matrix where you have N joints and M coordinates describing the

position and rotation of the foot].

If you hold your arm out straight in front of you with your palm

facing the ground and the axis of your elbow joint vertical you can

figure out a few elements of the kinematic jacobian of your hand

intuitively. Bending your elbow slightly moves your hand medially at

a rate equal to the length of your forearm times the velocity with

which you move your elbow joint. Rotating your shoulder about a

vertical axis at the same rate moves your hand in the same direction,

but about twice as fast, because your arm is about twice as long as

your forearm. Now, to get at the reason for having bent knees during

gait, try to move your hand directly towards your shoulder - doing

this requires moving the elbow and shoulder simultaneously in opposite

directions. If you move your hand at a constant velocity towards your

shoulder you'll see that the angular rate that you move your elbow at

slows down as your get closer to your shoulder. As you move further

from your shoulder, and closer to the 'kinematic singularity' that

occurs when you have a straight arm, the joint velocities needed to

keep your hand moving at a constant velocity increases. Analytically

there is an infinite vertical asymptote caused by dividing by a sin(

theta ) term as you approach the singular configuration - requiring

infinite joint velocity to maintain the same finite hand velocity.

> Would you say that the straight knee approach is another example of

> simplifying the human gait model. This model uses the momentum of the CoM

> Plus a small added energy input, in the robot case from electric motors,

> which is much more energy efficient than the Asimo model.

I think that's a fair assessment. There are unpowered passive dynamic

walkers (or more properly gravity powered passive dynamic walkers)

that have no motors or sensors and walk down slight inclines to

compensate for energy loss due to foot impact.

> Do you think a combination of the Asimo and the rocker foot robot would be a

> possibility that would be more analogous to 'normal or usual' human gait. EG

> it could recognise rough terrain or obstacles and converts from the

> efficient stiff knee rocker foot gait to the bent knee stable posture

> progression at the appropriate time.

There is definitely interest in merging the insights provided by these

two models. I think the key difference between them is that ASIMO and

kin have a stability metric that can be evaluated and enforced at

every instant in time, while the passive dynamic walkers use foot

placement and the dynamics of foot impact to stabilize and reject

disturbances over multiple steps. Your comments on energy storage in

human muscles and tendons remind me of the work being done by Hurst

and Grizzle on AMASC actuator based legs

[http://www.biped.ri.cmu.edu/] - that can efficiently store and return

mechanical energy.

Stuart