Contents
PR2 Gripper Transmission
Basic transmission equations:
![$$\theta = ACOS [ \frac{ a^2 + b^2 - ( L_0 + \frac{MR \times I}{GR})^2 - h^2}{2 a b} ] + \theta_0 - \phi_0$$](attachments/pr2_mechanism_model(2f)pr2_gripper_transmission/latex_b3c0ed6003531d322a11922f45b3b40315fc5956_p1.png "$$\theta = ACOS [ \frac{ a^2 + b^2 - ( L_0 + \frac{MR \times I}{GR})^2 - h^2}{2 a b} ] + \theta_0 - \phi_0$$")
pr2_gripper_transmission/latex_21fe9a050a799a72e4149902903091be83f1c1b6_p1.png "$$ t = 2 * (t_0 + r ( sin( \theta ) - sin( \theta_0 ) ) )$$")
Where
pr2_gripper_transmission/latex_b65922ff4dce6b30aaf4601f6c8a42e4f5400294_p1.png "$$t$$")
gripper gap size pr2_gripper_transmission/latex_ba277440d2351383d3360ac9d10291e7ebac1c8f_p1.png "$$MR$$")
motor revolution pr2_gripper_transmission/latex_caabaf5373caae61d46f0cd5b8ddebcc433afe20_p1.png "$$I$$")
screw drive pr2_gripper_transmission/latex_de6e7d4422e07c3de58211a8b59b0ba58b8ee20c_p1.png "$$\frac{mm}{rev}$$")
pr2_gripper_transmission/latex_8eb67a981a191b842b59ea0042ed0b18132d642b_p1.png "$$\theta$$")
intermediate variable. pr2_gripper_transmission/latex_6f27be11250279c2402dff602963a9d8eb86fca0_p1.png "$$GR$$")
gear ratio pr2_gripper_transmission/latex_bd370c23337526a0c79c1dcdef3efe115e88e904_p1.png "$$a,b,L_0,h,\theta_0,\phi_0,r$$")
geometric constants of the gripper
Jacobians
pr2_gripper_transmission/latex_03216ccc6d2f9f6745470460bf923e5c4c18790a_p1.png "$$ \frac{d \theta }{d MR} = \frac{ -1 }{ \sqrt{1 - u^2} } \frac{d u}{d MR} $$")
where
pr2_gripper_transmission/latex_f7762bc7155b9d79bfa29278afb1e390899ebc93_p1.png "$$u = \frac{ a^2 + b^2 - h^2 - ( L_0 + MR \frac{I}{GR} )^2 }{2 a b}$$")
pr2_gripper_transmission/latex_aea0de1509b62ff4e20108eae8b5879ea3d4f864_p1.png "$$\frac{d u}{d MR} = -\frac{d}{d MR} ( \frac{L_0^2}{2 a b} + MR \frac{L_0 I}{GR a b} + \frac{MR^2}{2ab} (\frac{I}{GR})^2 ) = - \frac{L_0 I}{GR a b} - \frac{MR}{ab} (\frac{I}{GR})^2 $$")
Inverse Transform (from $$\theta$$ to $$MR$$)
![$$ MR = \frac{GR}{I} [ -L_0 + \sqrt{ -(2 a b COS(\theta + \phi_0 - \theta_0) + h^2 - a^2 - b^2) } ]$$](attachments/pr2_mechanism_model(2f)pr2_gripper_transmission/latex_2be1d4395be9a8c926bebc9cc4c7a62b358e298b_p1.png "$$ MR = \frac{GR}{I} [ -L_0 + \sqrt{ -(2 a b COS(\theta + \phi_0 - \theta_0) + h^2 - a^2 - b^2) } ]$$")
pr2_gripper_transmission/latex_930db8332f65c562e130c198d8784aa6aa0c86d2_p1.png "$$\frac{d MR}{d \theta} = \frac{GR}{2 I} \frac{ 2 a b SIN(\theta + \phi_0 - \theta_0 )}{\sqrt{-(2 a b COS( \theta + \phi_0 - \theta_0) + h^2 - a^2 -b^2) }}$$")
![$$\theta = ASIN [ \frac{t - t_0}{r} + SIN( \theta_0) ]$$](attachments/pr2_mechanism_model(2f)pr2_gripper_transmission/latex_5da3dc5f31f1d888a6f5d2b7d717d7927c34038c_p1.png "$$\theta = ASIN [ \frac{t - t_0}{r} + SIN( \theta_0) ]$$")
pr2_gripper_transmission/latex_84ec4e68962345d42cf1b37e18e56759169cb199_p1.png "$$\frac{d \theta}{d t} = \frac{1}{\sqrt{ 1 - u^2 }} \frac{d u }{d t}$$")
where pr2_gripper_transmission/latex_75ed1e935bc5ca32e589dbb4add8cf23eff98962_p1.png "$$u = \frac{t - t_0}{r} + SIN( \theta_0) $$")
and pr2_gripper_transmission/latex_e02bc142baecb3271492def6415b82923b0c0362_p1.png "$$\frac{d u }{d t} = \frac{1}{r}$$")
Transmissions
Denote
joint_state->position by $$P_j$$ : gripper gap size in meters
actuator_state->position by $$P_a$$ : motor angle in radians
joint_state->velocity by $$V_j$$ : gripper gap velocity in meters per second
actuator_state->velocity by $$V_a$$ : motor angular rate in radians per second
joint_state->effort by $$E_j$$ : in Newtons
actuator_state->effort by $$E_a$$ : motor torque in Newton-Meters
Propagate From Actuator State to Joint State $$(P_a,V_a,E_a) \rarr (P_j,V_j,E_j)$$
Motor angle in radians ($$P_a$$) is converted to motor revolutions ($$MR$$),
pr2_gripper_transmission/latex_aa238b99b0fb1e567ef04e0e3ac0b42a08ce5840_p1.png "$$MR = \frac{P_a}{2 \pi}$$")
and we have the transmission equation for gripper gap size above
pr2_gripper_transmission/latex_452c1446dd74bed3e889f5776760bf9e1552edec_p1.png "$$P_j = t(\theta(MR))$$")
.
Motor angular rates in radians per second (pr2_gripper_transmission/latex_26db40fe8c6c6efbc6416b0fbe25d60a2e56a777_p1.png "$$V_a$$")
) is converted to revolutions per second,
pr2_gripper_transmission/latex_9ba320f8d002b89c0f380b57a8e41e7bdaca4230_p1.png "$$\dot{MR} = \frac{V_a}{2 \pi}$$")
and the gripper gap velocity is approximated by
pr2_gripper_transmission/latex_b431b57f16fc5e0af553ecadcc25b0b2db86a5da_p1.png "$$V_j = \dot{t} = \dot{MR} * \frac{ d t}{d MR}$$")
Given motor torque pr2_gripper_transmission/latex_4857ccacf41650ddbb3475dccebe77a13532a29d_p1.png "$$E_a$$")
, gripper effort pr2_gripper_transmission/latex_7e372dd5e3821406df87b9444bac8ddf5d90ae36_p1.png "$$E_j$$")
is approximated by
pr2_gripper_transmission/latex_e49932230491b14706029997a96bf5ce2fa58f8c_p1.png "$$E_j = E_a * \frac{d P_a}{d t} = I_m * \ddot{P_a} * \frac{d P_a}{d t} = I_m * \ddot{MR} * (2 \pi) * \frac{\frac{d MR}{d t}}{2 \pi} = MT * \frac{d MR}{d t}$$")
where pr2_gripper_transmission/latex_89aaa88153617f90db6b9fc58f473a84bbf756e0_p1.png "$$I_m$$")
is the moment of inertia at the motor and pr2_gripper_transmission/latex_1245fa4883eefb09e79cac0d1979ac0408b8f280_p1.png "$$MT$$")
is defined as pr2_gripper_transmission/latex_f05599fd34610976a31888cfdc0b4dc798bc0817_p1.png "$$I_m * \ddot{MR} = \frac{E_a}{2 \pi}$$")
is the motor torque in pr2_gripper_transmission/latex_b01271ec6041da4a9abae8483395d2d846312a83_p1.png "$$Kg * \frac{rev}{s^2}$$")
.
Propagate From Joint State to Actuator State $$(P_j,V_j,E_j) \rarr (P_a,V_a,E_a)$$
pretty much the reverse of previous section, given
pr2_gripper_transmission/latex_bafec3af28eab46edcbad62b181d0cff135c49e8_p1.png "$$t = P_j$$")
, inverse transmission gives pr2_gripper_transmission/latex_9cf47d0989c66d129fe1f0e480af489c41b51c26_p1.png "$$MR(\theta(t))$$")
, and
pr2_gripper_transmission/latex_580c8104951a9c451c6740e2234b30004583a754_p1.png "$$P_a = MR 2 \pi$$")
.
pr2_gripper_transmission/latex_f027c1aeb79565a3c012da38f609244e1237c1d4_p1.png "$$V_a = \dot{MR} 2 \pi = V_j * \frac{d MR}{d t} 2 \pi$$")
pr2_gripper_transmission/latex_fe03fb447192fbdd6ca2bb5df5bf972e56679b0a_p1.png "$$E_a = E_j \frac{d t}{d P_a} = E_j 2 \pi \frac{d t}{d MR}$$")
Known Limitations
- Gripper transmission produced gripper effort differs from actual measured gripper effort by +/-20%.