feat: add simple inverse-kinematics solver for multibodies (#632)

* feat: add a simple jacobian-based inverse-kinematics implementation for multibodies

* feat: add 2d inverse kinematics example

* feat: make forward_kinematics auto-fix the root’s degrees of freedom

* feat: add 3d inverse kinematics example

* chore: update changelog

* chore: clippy fixes

* chore: more clippy fixes

* fix tests

examples2d/inverse_kinematics2.rs

@@ -0,0 +1,96 @@
+use rapier2d::prelude::*;
+use rapier_testbed2d::Testbed;
+
+pub fn init_world(testbed: &mut Testbed) {
+    /*
+     * World
+     */
+    let mut bodies = RigidBodySet::new();
+    let mut colliders = ColliderSet::new();
+    let impulse_joints = ImpulseJointSet::new();
+    let mut multibody_joints = MultibodyJointSet::new();
+
+    /*
+     * Ground
+     */
+    let ground_size = 1.0;
+    let ground_height = 0.01;
+
+    let rigid_body = RigidBodyBuilder::fixed().translation(vector![0.0, -ground_height]);
+    let floor_handle = bodies.insert(rigid_body);
+    let collider = ColliderBuilder::cuboid(ground_size, ground_height);
+    colliders.insert_with_parent(collider, floor_handle, &mut bodies);
+
+    /*
+     * Setup groups
+     */
+    let num_segments = 10;
+    let body = RigidBodyBuilder::fixed();
+    let mut last_body = bodies.insert(body);
+    let mut last_link = MultibodyJointHandle::invalid();
+
+    for i in 0..num_segments {
+        let size = 1.0 / num_segments as f32;
+        let body = RigidBodyBuilder::dynamic().can_sleep(false);
+        let new_body = bodies.insert(body);
+        // NOTE: we add a sensor collider just to make the destbed draw a rectangle to make
+        //       the demo look nicer. IK could be used without cuboid.
+        let collider = ColliderBuilder::cuboid(size / 8.0, size / 2.0)
+            .density(0.0)
+            .sensor(true);
+        colliders.insert_with_parent(collider, new_body, &mut bodies);
+
+        let link_ab = RevoluteJointBuilder::new()
+            .local_anchor1(point![0.0, size / 2.0 * (i != 0) as usize as f32])
+            .local_anchor2(point![0.0, -size / 2.0])
+            .build()
+            .data;
+
+        last_link = multibody_joints
+            .insert(last_body, new_body, link_ab, true)
+            .unwrap();
+
+        last_body = new_body;
+    }
+
+    let mut displacements = DVector::zeros(0);
+
+    testbed.add_callback(move |graphics, physics, _, _| {
+        let Some(graphics) = graphics else { return };
+        if let Some((multibody, link_id)) = physics.multibody_joints.get_mut(last_link) {
+            // Ensure our displacement vector has the right number of elements.
+            if displacements.nrows() < multibody.ndofs() {
+                displacements = DVector::zeros(multibody.ndofs());
+            } else {
+                displacements.fill(0.0);
+            }
+
+            let Some(mouse_point) = graphics.mouse().point else {
+                return;
+            };
+
+            // We will have the endpoint track the mouse position.
+            let target_point = mouse_point.coords;
+
+            let options = InverseKinematicsOption {
+                constrained_axes: JointAxesMask::LIN_AXES,
+                ..Default::default()
+            };
+
+            multibody.inverse_kinematics(
+                &physics.bodies,
+                link_id,
+                &options,
+                &Isometry::from(target_point),
+                &mut displacements,
+            );
+            multibody.apply_displacements(displacements.as_slice());
+        }
+    });
+
+    /*
+     * Set up the testbed.
+     */
+    testbed.set_world(bodies, colliders, impulse_joints, multibody_joints);
+    testbed.look_at(point![0.0, 0.0], 300.0);
+}