Added an illustrative example to demonstrate the workflow of

pydy_viz

illustrative_example_for_viz/illustrative_example.py

@@ -0,0 +1,221 @@
+#This program represents a hypothetical situation for a complete workflow
+#for simulating a three link pendulum with links as rigid bodies
+from sympy import symbols,sympify
+from sympy.physics.mechanics import *
+
+#Number of links = 3
+N_links = 3
+
+#Number of masses = 3
+N_bobs = 3
+
+#Defining Dynamic Symbols ................
+
+#Generalized coordinates(angular) ...
+
+alpha = dynamicsymbols('alpha1 alpha2 alpha3')
+beta = dynamicsymbols('beta1 beta2 beta3')    
+
+#Generalized speeds(angular) ...
+alphad = dynamicsymbols('alpha1 alpha2 alpha3',1)
+betad = dynamicsymbols('beta1 beta2 beta3',1)
+
+#Mass of each bob:
+m = symbols('m:'+str(N_bobs))
+
+
+#Length mass and radii of each link(assuming as rods) ..
+l = symbols('l:' + str(N_links)) 
+M = symbols('M:' + str(N_links)) 
+radii = symbols('radii:' + str(N_links)) 
+
+#For storing Inertia for each link :
+Ixx = symbols('Ixx:'+str(N_links))
+Iyy = symbols('Iyy:'+str(N_links))
+
+#gravity and time ....
+g, t = symbols('g t')
+
+
+#Now defining an Inertial ReferenceFrame First ....
+
+I = ReferenceFrame('I')
+
+#And some other frames ...
+
+A = ReferenceFrame('A')
+A.orient(I,'Body',[alpha[0],beta[0],0],'ZXY')
+B = ReferenceFrame('B')
+B.orient(I,'Body',[alpha[1],beta[1],0],'ZXY')
+C = ReferenceFrame('C')
+C.orient(I,'Body',[alpha[2],beta[2],0],'ZXY')
+
+
+#Setting angular velocities of new frames ...
+A.set_ang_vel(I, alphad[0] * I.z + betad[0] * I.x)
+B.set_ang_vel(I, alphad[1] * I.z + betad[1] * I.x)
+C.set_ang_vel(I, alphad[2] * I.z + betad[2] * I.x)
+
+
+
+# An Origin point, with velocity = 0
+O = Point('O')
+O.set_vel(I,0)
+
+#Three more points, for masses ..
+P1 = O.locatenew('P1', l[0] * A.y)
+P2 = O.locatenew('P2', l[1] * B.y)
+P3 = O.locatenew('P3', l[2] * C.y)
+
+#Setting velocities of points with v2pt theory ...
+P1.v2pt_theory(O, I, A)
+P2.v2pt_theory(P1, I, B)
+P3.v2pt_theory(P2, I, C)
+points = [P1,P2,P3]
+
+Pa1 = Particle('Pa1', points[0], m[0])
+Pa2 = Particle('Pa2', points[1], m[1])
+Pa3 = Particle('Pa3', points[2], m[2])
+particles = [Pa1,Pa2,Pa3]
+
+
+
+#defining points for links(RigidBodies)
+#Assuming CoM as l/2 ...
+P_link1 = O.locatenew('P_link1', l[0]/2 * A.y)
+P_link2 = O.locatenew('P_link1', l[1]/2 * B.y)
+P_link3 = O.locatenew('P_link1', l[2]/2 * C.y)
+
+#setting velocities of these points with v2pt theory ...
+P_link1.v2pt_theory(O, I, A)
+P_link2.v2pt_theory(P_link1, I, B)
+P_link3.v2pt_theory(P_link2, I, C)
+
+points_rigid_body = [P_link1,P_link2,P_link3]
+
+
+#defining inertia tensors for links
+
+inertia_link1 = inertia(A,Ixx[0],Iyy[0],0)
+inertia_link2 = inertia(B,Ixx[1],Iyy[1],0)
+inertia_link3 = inertia(C,Ixx[2],Iyy[2],0)
+
+#Defining links as Rigid bodies ...
+
+link1 = RigidBody('link1', P_link1, A, M[0], (inertia_link1, O))
+link2 = RigidBody('link2', P_link2, B, M[1], (inertia_link2, P1))
+link3 = RigidBody('link3', P_link3, C, M[2], (inertia_link3, P2))
+links = [link1,link2,link3]
+
+
+#Defining a basic shape for links ..
+rod1 = Cylinder(length=l[0],radii=radii[0])
+rod2 = Cylinder(length=l[1],radii=radii[1])
+rod3 = Cylinder(length=l[2],radii=radii[2])
+
+link1.shape(rod1)
+link2.shape(rod2)
+link3.shape(rod3)
+
+#Applying forces on all particles , and adding all forces in a list..
+forces = []
+for particle in particles:
+
+    mass =  particle.get_mass()
+    point =  particle.get_point()
+    forces.append((point, -mass * g * I.y) )
+
+#Applying forces on rigidbodies ..
+for link in links:
+    mass = link.get_mass()
+    point = link.get_masscenter()
+    forces.append((point, -mass * g * I.y) ) 
+
+kinetic_differentials = []
+for i in range(0,N_bobs):
+    kinetic_differentials.append(alphad[i] - alpha[i])
+    kinetic_differentials.append(betad[i] - beta[i])
+
+#Adding particles and links in the same system ...
+total_system = []
+for particle in particles:
+    total_system.append(particle)
+
+for link in links:
+    total_system.append(link)
+
+q = []
+for angle in alpha:
+	q.append(angle)
+for angle in beta:
+	q.append(angle)
+print q
+u = []
+
+for vel in alphad:
+	u.append(vel)
+for vel in betad:
+	u.append(vel)
+
+print u		
+kane = KanesMethod(I, q_ind=q, u_ind=u, kd_eqs=kinetic_differentials)
+fr, frstar = kane.kanes_equations(forces, total_system)
+
+print fr
+
+#Now we have symbolic equations of motion. ..
+# we integrate them numerically. ..
+
+params = [g ,l1,l2,l3,m1,m2,m3,M1,M2,M3]
+
+param_vals = [9.8 ,1.0,1.0,1.0,2,2,2,5,5,5]
+
+right_hand_side = code_generator(kane,params)   
+
+#setting initial conditions ..
+init_conditions = [radians(45),radians(45),radians(30),\
+                   radians(30),radians(15),radians(15),\
+                           0,           0,         0,\
+                           0,           0,          0]
+t = [0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0]
+
+numerical_vals = odeint(right_hand_side,init_conditions,t)
+
+#Now for each t, we have numerical vals of coordinates ..
+#Now we set up a visualization frame, 
+
+frame1 = VisualizationFrame('frame1',I,O)
+
+frame1.add_rigidbodies(links)
+
+frame1.add_particles(particles)
+
+param_vals_for_viz = {'g':9.8 ,'l1':1.0,'l2':1.0,'l3':1.0,'m1':2,'m2':2,'m3':2,'M1':5,'M1':5,'M1':5]
+
+json = frame1.generate_json(initial_conditions,q)
+#Here we can replace initial_conditions with the conditions at any 
+#specific time interval ....
+
+##This line does following things ...
+##calls RigidBody.transform_matrix() on all rigidbodies, 
+#so that we have info of rigidbodies(CoM,rotation,translation)
+# w.r.t VisualizationFrame('I' in this case) 
+##Even if they are defined in any other frame ....
+##calls point.set_pos() for all particles with arg 
+##as Point in VisualizationFrame(O here) ..
+##With these and init_conditions, we have a starting point of visualization ..
+
+scene = Scene()
+Scene.view(json)      # Just visualize/view the system at initial_conditions,
+                      #w.r.t VIsualizationFrame(I in this case) 
+                      # No simulation here. ..
+
+Scene.simulate(json,numerical_vals)  # modify the input json,
+									 #Add the values at different times from numerical_vals,
+									 #To the json, which is then passed to 
+									 #javascript
+									 #
+									 #(alpha1,alpha2,alpha3,beta1,beta2,beta3) at time=t 						 	
+
+
+

illustrative_example_for_viz/three_link_pendulum.py

@@ -0,0 +1,157 @@
+#For this one we assume RigidBody links
+from sympy import symbols,sympify
+from sympy.physics.mechanics import *
+
+#Number of links = 3
+N_links = 3
+
+#Number of masses = 3
+N_bobs = 3
+
+#Defining Dynamic Symbols ................
+
+#Generalized coordinates(angular) ...
+
+alpha = dynamicsymbols('alpha1 alpha2 alpha3')
+beta = dynamicsymbols('beta1 beta2 beta3')    
+
+#Generalized speeds(angular) ...
+alphad = dynamicsymbols('alpha1 alpha2 alpha3',1)
+betad = dynamicsymbols('beta1 beta2 beta3',1)
+
+#Mass of each bob:
+m = symbols('m:'+str(N_bobs))
+
+
+#Length and mass of each link ..
+l = symbols('l:' + str(N_links)) 
+M = symbols('M:' + str(N_links)) 
+#For storing Inertia for each link :
+Ixx = symbols('Ixx:'+str(N_links))
+Iyy = symbols('Iyy:'+str(N_links))
+
+#gravity and time ....
+g, t = symbols('g t')
+
+
+#Now defining an Inertial ReferenceFrame First ....
+
+I = ReferenceFrame('I')
+
+#And some other frames ...
+
+A = ReferenceFrame('A')
+A.orient(I,'Body',[alpha[0],beta[0],0],'ZXY')
+B = ReferenceFrame('B')
+B.orient(I,'Body',[alpha[1],beta[1],0],'ZXY')
+C = ReferenceFrame('C')
+C.orient(I,'Body',[alpha[2],beta[2],0],'ZXY')
+
+
+#Setting angular velocities of new frames ...
+A.set_ang_vel(I, alphad[0] * I.z + betad[0] * I.x)
+B.set_ang_vel(I, alphad[1] * I.z + betad[1] * I.x)
+C.set_ang_vel(I, alphad[2] * I.z + betad[2] * I.x)
+
+
+
+# An Origin point, with velocity = 0
+O = Point('O')
+O.set_vel(I,0)
+
+#Three more points, for masses ..
+P1 = O.locatenew('P1', l[0] * A.y)
+P2 = O.locatenew('P2', l[1] * B.y)
+P3 = O.locatenew('P3', l[2] * C.y)
+
+#Setting velocities of points with v2pt theory ...
+P1.v2pt_theory(O, I, A)
+P2.v2pt_theory(P1, I, B)
+P3.v2pt_theory(P2, I, C)
+points = [P1,P2,P3]
+
+Pa1 = Particle('Pa1', points[0], m[0])
+Pa2 = Particle('Pa2', points[1], m[1])
+Pa3 = Particle('Pa3', points[2], m[2])
+particles = [Pa1,Pa2,Pa3]
+
+
+
+#defining points for links(RigidBodies)
+#Assuming CoM as l/2 ...
+P_link1 = O.locatenew('P_link1', l[0]/2 * A.y)
+P_link2 = O.locatenew('P_link1', l[1]/2 * B.y)
+P_link3 = O.locatenew('P_link1', l[2]/2 * C.y)
+#setting velocities of these points with v2pt theory ...
+P_link1.v2pt_theory(O, I, A)
+P_link2.v2pt_theory(P_link1, I, B)
+P_link3.v2pt_theory(P_link2, I, C)
+
+points_rigid_body = [P_link1,P_link2,P_link3]
+
+
+#defining inertia tensors for links
+
+inertia_link1 = inertia(A,Ixx[0],Iyy[0],0)
+inertia_link2 = inertia(B,Ixx[1],Iyy[1],0)
+inertia_link3 = inertia(C,Ixx[2],Iyy[2],0)
+
+#Defining links as Rigid bodies ...
+
+link1 = RigidBody('link1', P_link1, A, M[0], (inertia_link1, O))
+link2 = RigidBody('link2', P_link2, B, M[1], (inertia_link2, P1))
+link3 = RigidBody('link3', P_link3, C, M[2], (inertia_link3, P2))
+links = [link1,link2,link3]
+
+
+#Applying forces on all particles , and adding all forces in a list..
+forces = []
+for particle in particles:
+
+    mass =  particle.get_mass()
+    point =  particle.get_point()
+    forces.append((point, -mass * g * I.y) )
+
+#Applying forces on rigidbodies ..
+for link in links:
+    mass = link.get_mass()
+    point = link.get_masscenter()
+    forces.append((point, -mass * g * I.y) ) 
+kinetic_differentials = []
+for i in range(0,N_bobs):
+    kinetic_differentials.append(alphad[i] - alpha[i])
+    kinetic_differentials.append(betad[i] - beta[i])
+
+#Adding particles and links in the same system ...
+total_system = []
+for particle in particles:
+    total_system.append(particle)
+
+for link in links:
+    total_system.append(link)
+
+q = []
+for angle in alpha:
+	q.append(angle)
+for angle in beta:
+	q.append(angle)
+print q
+u = []
+
+for vel in alphad:
+	u.append(vel)
+for vel in betad:
+	u.append(vel)
+
+print u		
+kane = KanesMethod(I, q_ind=q, u_ind=u, kd_eqs=kinetic_differentials)
+fr, frstar = kane.kanes_equations(forces, total_system)
+
+print fr
+
+
+
+
+
+
+

three_link_pendulum/three_link_pendulum.py

@@ -148,10 +148,3 @@
 fr, frstar = kane.kanes_equations(forces, total_system)
 
 print fr
-
-
-
-
-
-
-