Verify the properties of skew symmetry, passivity and linearity in the parameters for the system given by Equation (9.6). Compute bounds on the inertia matrix, M(q), in terms of bounds on D(q). Show that M(g) is positive definite. 9-1 Verify the properties of skew symmetry, passivity and linearity in the parameters for the system given by solutionspile.com

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Verify the properties of skew symmetry, passivity and linearity in the parameters for the system given by Equation (9.6). Compute bounds on the inertia matrix, M(q), in terms of bounds on D(q). Show that M(g) is positive definite.

9-1 Verify the properties of skew symmetry, passivity and linearity in the parameters for the system given by Equation (9.6). Compute bounds on the inertia matrix,

M (q)

, in terms of bounds on

D (q)

. Show that

M (q)

is positive definite. for

k = 1, \dots, n

, where

B_{k} = B_{m_{k}} + K_{b_{k}} K_{m_{k}} / R_{k}

. Multiplying Equation (9.2) by the gear ratio

r_{k}

and using the fact that the motor angles

?_{m_{k}}

and the link angles

q_{k}

are related by

?_{m_{k}} = r_{k} q_{k}

we may write Equation (9.2) as

r_{k}^{2} J_{m} \overset{q}{¨}_{k} + r_{k}^{2} B_{k} \overset{q}{?}_{k} = r_{k} K_{m_{k}} / R V_{k} ? ?_{k}

Substituting Equation (9.4) into Equation (9.1) yields

r_{k}^{2} J_{m_{k}} \overset{q}{¨}_{k} + ?_{j = 1}^{n} d_{kj} \overset{q}{¨}_{j} + ?_{i, j = 1}^{n} c_{ijk} \overset{q}{?}_{i} \overset{q}{?}_{j} + r_{k}^{2} B_{k} \overset{q}{?}_{k} + g_{k} = r_{k} \frac{K _{m}}{R} V_{k}

for

k = 1, \dots, n

. In matrix form Equation (9.5) can be written as

M (q) \overset{q}{¨} + C (q, \overset{q}{?}) \overset{q}{?} + B \overset{q}{?} + g (q) = u

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