Ball-Balancing-Robot/robotKinematics.py at main · I-M-Robotics-Lab/Ball-Balancing-Robot · GitHub

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import math
import numpy as np


class RobotKinematics:

    def __init__(self, lp=7.125, l1=6.20, l2=4.50, lb=4.00, invert=False):

        self.lp = lp    #Radius of Top
        self.l1 = l1    #Top Arm
        self.l2 = l2    #Bottom Arm
        self.lb = lb    #Radius of Bottom
        self.invert = invert    #Whether the arm stays inward or outward

        self.maxh = self.compute_maxh() - 0.2    #maximum height that the Top plane should be
        self.minh = self.compute_minh() + 0.45
        self.p = [0.0,0.0,self.maxh]    #Center of the Top plane
        self.h = (self.maxh + self.minh)/2
        self.maxtheta = 10

        #Top Nodes
        self.A1 = [0,0,0]
        self.A2 = [0,0,0]
        self.A3 = [0,0,0]

        #Bottom Nodes
        self.B1 = [0,0,0]
        self.B2 = [0,0,0]
        self.B3 = [0,0,0]

        #Middle Nodes
        self.C1 = [0.0, 0.0, 0.0]
        self.C2 = [0.0, 0.0, 0.0]
        self.C3 = [0.0, 0.0, 0.0]

        self.max_theta(self.h)

        self.theta1 = 0
        self.theta2 = 0
        self.theta3 = 0

    def compute_maxh(self):
        return math.sqrt(((self.l1 + self.l2) ** 2) - ((self.lp - self.lb) ** 2))

    def compute_minh(self):
        if self.l1 > self.l2:
            return math.sqrt((self.l1 ** 2) - ((self.lb + self.l2 - self.lp) ** 2))
        elif self.l2 > self.l1:
            return math.sqrt(((self.l2 - self.l1) ** 2) - ((self.lp - self.lb) ** 2))
        else:
            return 0

    def solve_top(self, a, b, c, h): #Orientation vector n: [alpha, beta, gamma], h: height

        if not self.invert:
            #A1, A2, A3 are ball-joint coordinates, A2 is the vertex on plane y=0
            self.A1 = [ -(self.lp*c) / (math.sqrt(4*c**2 + (a - math.sqrt(3)*b)**2)),
                (math.sqrt(3)*self.lp*c) / (math.sqrt(4*c**2 + (a - math.sqrt(3)*b)**2)),
                h + ((a - math.sqrt(3)*b)*self.lp) / (math.sqrt(4*c**2 + (a - math.sqrt(3)*b)**2))]

            self.A2 = [ (self.lp*c) / (math.sqrt(c**2 + a**2)),
                    0,
                        h - ((self.lp*a) / (math.sqrt(c**2 + a**2)))]

            self.A3 = [ -(self.lp*c) / (math.sqrt(4*c**2 + (a + math.sqrt(3)*b)**2)),
                -(math.sqrt(3)*self.lp*c) / (math.sqrt(4*c**2 + (a + math.sqrt(3)*b)**2)),
                h + ((a + math.sqrt(3)*b)*self.lp) / (math.sqrt(4*c**2 + (a + math.sqrt(3)*b)**2))]
        else:
            self.A1 = [ -(self.lp*c) / (math.sqrt(4*c**2 + (a - math.sqrt(3)*b)**2)),
                (math.sqrt(3)*self.lp*c) / (math.sqrt(4*c**2 + (a - math.sqrt(3)*b)**2)),
                h + ((a - math.sqrt(3)*b)*self.lp) / (math.sqrt(4*c**2 + (a - math.sqrt(3)*b)**2))]

            self.A2 = [ (self.lp*c) / (math.sqrt(c**2 + a**2)),
                    0,
                        h - ((self.lp*a) / (math.sqrt(c**2 + a**2)))]

            self.A3 = [ -(self.lp*c) / (math.sqrt(4*c**2 + (a + math.sqrt(3)*b)**2)),
                -(math.sqrt(3)*self.lp*c) / (math.sqrt(4*c**2 + (a + math.sqrt(3)*b)**2)),
                h + ((a + math.sqrt(3)*b)*self.lp) / (math.sqrt(4*c**2 + (a + math.sqrt(3)*b)**2))]


    def solve_middle(self):


        a11, a12, a13 = map(float, self.A1)
        a21, a22, a23 = map(float, self.A2)
        a31, a32, a33 = map(float, self.A3)


        p1 = (-a11 + math.sqrt(3)*a12 - 2*self.lb) / a13
        q1 = (a11**2 + a12**2 + a13**2 + self.l2**2 - self.l1**2 - self.lb**2) / (2*a13)
        r1 = p1**2 + 4
        s1 = 2*p1*q1 + 4*self.lb
        t1 = q1**2 + self.lb**2 - self.l2**2

        p2 = (self.lb - a21) / a23
        q2 = (a21**2 + a23**2 - self.lb**2 + self.l2**2 - self.l1**2) / (2*a23)
        r2 = p2**2 + 1
        s2 = 2*(p2*q2 - self.lb)
        t2 = q2**2 - self.l2**2 + self.lb**2

        p3 = (-a31 - math.sqrt(3)*a32 - 2*self.lb) / a33
        q3 = (a31**2 + a32**2 + a33**2 + self.l2**2 - self.l1**2 - self.lb**2) / (2*a33)
        r3 = p3**2 + 4
        s3 = 2*p3*q3 + 4*self.lb
        t3 = q3**2 + self.lb**2 - self.l2**2

        if not self.invert:

            c11 = (-s1 - math.sqrt(s1**2 - 4*r1*t1)) / (2*r1)
            c12 = -math.sqrt(3) * c11
            c13 = math.sqrt(self.l2**2 - 4*(c11**2) - 4*self.lb*c11 - self.lb**2)

            self.C1 = [c11, c12, c13]

            c21 = (-s2 + math.sqrt(s2**2 - 4*r2*t2)) / (2*r2)
            c22 = 0
            c23 = math.sqrt(self.l2**2 - (c21 - self.lb)**2)

            self.C2 = [c21, c22, c23]

            c31 = (-s3 - math.sqrt(s3**2 - 4*r3*t3)) / (2*r3)
            c32 =  math.sqrt(3) * c31
            c33 = math.sqrt(self.l2**2 - 4*(c31**2) - 4*self.lb*c31 - self.lb**2)

            self.C3 = [c31, c32, c33]

        else:

            c11 = (-s1 - math.sqrt(s1**2 - 4*r1*t1)) / (2*r1)
            c12 = -math.sqrt(3) * c11
            c13 = math.sqrt(self.l2**2 - 4*(c11**2) - 4*self.lb*c11 - self.lb**2)

            self.C1 = [c11, c12, -c13]

            c21 = (-s2 + math.sqrt(s2**2 - 4*r2*t2)) / (2*r2)
            c22 = 0
            c23 = math.sqrt(self.l2**2 - (c21 - self.lb)**2)

            self.C2 = [c21, c22, -c23]

            c31 = (-s3 - math.sqrt(s3**2 - 4*r3*t3)) / (2*r3)
            c32 = math.sqrt(3) * c31
            c33 = math.sqrt(self.l2**2 - 4*(c31**2) - 4*self.lb*c31 - self.lb**2)

            self.C3 = [c31, c32, -c33]

    def solve_inverse_kinematics_vector(self, a, b, c, h):

        self.B1 = [-0.5*self.lb, math.sqrt(3)*0.5*self.lb,0]
        self.B2 = [self.lb,0,0]
        self.B3 = [-0.5*self.lb, -1*math.sqrt(3)*0.5*self.lb,0]

        self.solve_top(a, b, c, h)
        self.solve_middle()

        self.theta1 = math.pi/2 - math.atan2(math.sqrt(self.C1[0]**2 + self.C1[1]**2) - self.lb, self.C1[2])
        self.theta2 = math.atan2(self.C2[2], self.C2[0] - self.lb)
        self.theta3 = math.pi/2 - math.atan2(math.sqrt(self.C3[0]**2 + self.C3[1]**2) - self.lb, self.C3[2])

    def solve_inverse_kinematics_spherical(self, theta, phi, h): #psi = azimuthal angle, theta = polar angle

        #conversion
        self.h = h
        self.max_theta(h)

        theta = min(theta, self.maxtheta)

        a = math.sin(math.radians(theta)) * math.cos(math.radians(phi))
        b  =  math.sin(math.radians(theta)) * math.sin(math.radians(phi))
        c = math.cos(math.radians(theta))

        try:
            self.solve_inverse_kinematics_vector(a,b,c,h)
        except Exception as e:
            print(a,b,c,h, theta, phi)
            pass


    def max_theta(self, h, tol=1e-3):
        theta_low, theta_high = 0.0, math.radians(20)
        def valid(theta):
            c = math.cos(theta)
            for s in (1, -1):
                a21 = self.lp * c
                a23 = h - self.lp * (s * math.sin(theta))
                try:
                    p2 = (self.lb - a21) / a23
                    q2 = (a21**2 + a23**2 - self.lb**2 + self.l2**2 - self.l1**2) / (2 * a23)
                    r2 = p2**2 + 1
                    s2 = 2 * (p2 * q2 - self.lb)
                    t2 = q2**2 - self.l2**2 + self.lb**2
                    disc = s2**2 - 4 * r2 * t2
                    if disc < 0: return False
                    c21 = (-s2 + math.sqrt(disc)) / (2 * r2)
                    delta = self.l2**2 - (c21 - self.lb)**2
                    if delta < 0: return False
                    c23 = math.sqrt(delta)
                    if abs(math.sqrt((a21-c21)**2 + (a23-c23)**2) - self.l1) > 1e-3: return False
                    if abs(math.sqrt((self.lb-c21)**2 + c23**2) - self.l2) > 1e-3: return False
                except:
                    return False
            return True
        while theta_high - theta_low > tol:
            theta_mid = (theta_low + theta_high) / 2
            if valid(theta_mid): theta_low = theta_mid
            else: theta_high = theta_mid

        self.maxtheta = max(0, math.degrees(round(theta_low, 4)) - 0.5)