计算/近似刚体上的速度和旋转冲量，只给出一个位置上的力和刚体的质心？

Question

// The velocity of the particle at the time of contact
float4 contactVelocity;

// The index of the shape that the particle collided with
int shapeIndex;

// The particle's position in the world (which should be the same as the contact point)
float3 pos;

// The mass of the particle
float mass;

// The shape's center of mass
float3 shapeOrigin;

// TODO: define ApplyVelForce & ApplyRotForce
velDelta[shapeIndex] = velDelta[shapeIndex] + GetVelForce(shapeOrigin, contactVelocity * mass, pos);
rotVelDelta[shapeIndex] = rotVelDelta[shapeIndex] + GetRotForce(shapeOrigin, contactVelocity * mass, pos);

// function definitions
float3 GetVelForce(float3 shapeCentroid, float3 force, float3 forcePoint){ /* TODO define */ }
float3 GetRotForce(float3 shapeCentroid, float3 force, float3 forcePoint){ /* TODO define */ }

如果有人知道一个相对简单的公式来合理高效地计算甚至近似这些速度和旋转力，请告诉我。我在谷歌搜索了一下，但所有关于这方面的文章似乎都超出了我的理解范围。我只是不认为我有足够的经验和知识的运动学，以计算公式，像这些我自己。

Answer 1

你想要的东西通常是不确定的，给出了你想要的信息作为输入。在一般情况下，您需要作为输入：

Input:

center of mass of shape : x_c (changes with time)
orientation of shape in space : U (orthogonal matrix, equivalent to a coordinate frame
                                    attached to and moving with the shape, 
                                    changes with time)
Force vector in world space : f (changes with time)
point of force in world space : x_F (changes with time)
inertia matrix of the shape : J (constant) 
mass of shape : m (constant)

Output:
Velocity of center of mass : v_c (changes with time)
Angular velocity of shape in world frame : o (changes with time)

一般而言，刚体在任何给定时刻在世界框架中的位置和方位的完整描述如下：

x_c = x_c(t), U = U(t)

其中u为3×3正交矩阵，其列是与刚体牢固附着并随刚体运动的坐标框架的3个单位向量的坐标，原点为x_c点。

point of force: 
x_F = x_c + U * X_F

force:
f = U*F

angular velocity (along the momentary axis of rotation):
o = U*O

运动方程是

d/dt(x_c) = v_c

d/dt(v_c) = U*F / m

d/dt (J*O) = cross(J*O, O) + U.T * cross(X_F, F)

d/dt U = U*matrix(O)

哪里

x_c = center of mass,  
v_c = velocity of center of mass,  
O = angular velocity in body fixed frame,  
U = 3 by 3 orthogonal a matrix with columns the 
    orientations of the axis of the body fixed system 
F = force in body-fixed frame
X_F = point of force in body fixed frame

因此，产生刚体运动的一个简单的迭代格式是

input:
x_c_in, v_c_in, U_in, O_in, J, m, X_F, F, h = time-step 

x_c_out = x_c_in + h*v_c_in
v_c_out = v_c_in + h*(U_in * F_in)/m
Rot_hO = rotate(h*O_in) 
        (perform rotation around axis determined by vector O and angle h*norm(O))
U_out = U_in * Rot_hO
O_out = O_in + h*inverse(J)*( cross(J*O_in, O_in) + cross(X_F, F) )

output:
x_c_out, v_c_out, U_out, O_out,