Slip use = tan(θ−ε)/μs, tip use = h·tan(θ−ε)/d. A bar crossing 1.0 means failure.
The science
Friction cone. A hoof stays put while the tangential force it needs is at most the static friction limit, f ≤ μs·N. On a slope of angle θ the weight splits into N = m g cosθ and f = m g sinθ, so standing is possible while tanθ ≤ μs, i.e. θslip = arctan μs. The mass cancels: a kid and a billy slip at the same angle.
Footholds. If the hoof rests on a small ledge tilted by ε into the slope, the contact only feels the local angle θ−ε, so θslip = arctan μs + ε. This is why goats pick ledges, cracks and steps. Once a hoof starts skidding it leaves the ledge, so sliding is modelled on the smooth slab.
Sliding. With kinetic friction μk (assumed 0.85 μs) the body decelerates or accelerates down the slope with a = g(sinθ − μk cosθ), so s = ½ a t² from rest. The checks panel integrates the motion numerically and compares it with this formula.
Tip-over. A rigid animal topples about its downhill hooves when the vertical line through the centre of mass leaves the support footprint: h·tan(θ−ε) > d, with h the centre-of-mass height and d its distance to the downhill hooves along the slope. Crouching lowers h and, with the legs splayed, increases d. Across the slope d is only half the track width, which is why traversing needs ledges. Once the torque τ = m g (r_y sinθ − r_x cosθ) is positive the body rotates as a rigid pendulum with I φ'' = τ, I = m(k²+d²+h²).
Cost of climbing. Mechanical power is P = m g v sinθ; with a muscle efficiency of about 25 % (typical value) the metabolic power is P/0.25.
Honest simplifications. The goat is one rigid body with a fixed walking animation; real goats shift weight, brake with their dew claws, have a soft rubbery hoof pad plus a hard rim and use dynamic gaits. The μ values are typical illustrative numbers, not measurements of a specific rock. A smooth slab is a worst case: the real cliff has the footholds that this model lets you add with ε. Starting a climb adds the acceleration term a/(g cosθ) to the friction demand.