Table 4.

Nomenclature used in equations of motion.

${\hat{a}}^k$	linear acceleration of the centre of mass of segment k
${\hat{c}}^k$	vector from centre of rotation of joint at proximal end of segment k to centre of mass of segment k
${\hat{d}}^k$	vector from centre of rotation of joint at proximal end of segment k to centre of rotation joint at distal end of segment k
${\tilde{d}}^k$	skew-symmetric matrix of vector
${\tilde{d}}_l^3$	skew-symmetric matrix of vector from centre of rotation of hip to tibiofemoral joint contact l
E3×3	3×3 matrix of zeros
${\tilde{f}}^3$	skew-symmetric matrix of vector from centre of rotation of hip to contact point of patella with the femur
Fi	magnitude of force in muscle i
$F_{{max}_i}$	maximum possible force in muscle i (upper bound)
$\hat{g}$	acceleration due to gravity
${\tilde{h}}_l^2$	skew-symmetric matrix of vector from centre of rotation of knee to tibiofemoral joint contact l
i	muscle number
I3×3	3×3 identity matrix
j	ligament number
J	cost function
k	segment number
Lj	magnitude of force in ligament j
$L_{\underset{j}{max}}$	maximum possible force in ligament j (upper bound)
mk	mass of segment k
M	total number of muscles
N	total number of ligaments
${\hat{p}}_i^k$	unit vector representing the line of action of force created by muscle i that acts on segment k (zero if muscle does not insert on segment k)
pat	patella
pt	patellar tendon
P/Q ratio	ratio of patellar tendon to quadriceps tendon force
${\hat{q}}_j^k$	unit vector representing the line of action of force created by ligament j that acts on segment k (zero if ligament does not insert on segment k)
${\hat{r}}_i^k$	vector from centre of rotation of joint at proximal end of segment k to point of action of muscle i on segment k (zero if muscle does not insert on segment k)
Rxk	x component of reaction force acting at proximal end of segment k
Ryk	y component of reaction force acting at proximal end of segment k
Rzk	z component of reaction force acting at proximal end of segment k
${\hat{R}}^k$	vector representing x, y and z components of reaction force acting at proximal end of segment k
${\hat{R}}_l^k$	vector representing x, y and z components of reaction force l acting at proximal end of segment k
${\hat{s}}_j^k$	vector from centre of rotation of joint at proximal end of segment k to point of action of ligament j on segment k (zero if ligament does not insert on segment k)
${-\hat{S}}^k$	inter-segmental force acting on proximal end of segment k
${\hat{v}}_i^k$	effective moment arm of muscle i on segment k
${\hat{w}}_j^k$	effective moment arm of ligament j on segment k
${-\hat{W}}^k$	inter-segmental moment acting on proximal end of segment k
$Y_{3\times 3}^k$	inertia tensor of segment k
ρi	P/Q ratio for muscle i (zero if the muscle is not part of the quadriceps muscle group)
${\dot{\hat{\phi }}}^k$	angular velocity of segment k
${\ddot{\hat{\phi }}}^k$	angular acceleration of segment k